SAT Geometry and Trigonometry practice questions
Angles, triangles, circles, volume and right-triangle trigonometry, usually with a diagram to decode.
685 free questions, each with a full worked solution and an explanation of what each wrong answer means.
Practise this domain adaptively →Easy (136)
- Triangle angle sum
Two angles of a triangle measure 40° and 75°. What is the measure of the third angle? - Area of a triangle
A triangle has a base of 10 and a height of 6. What is its area? - Special right triangles (30-60-90)
In a 30°-60°-90° right triangle, the shorter leg (opposite the 30° angle) has length 6. What is the length of the hypote… - Perimeter of a rectangle
A rectangle has a length of 8 and a width of 3. What is its perimeter? - Volume of a rectangular prism
A closed rectangular box has a length of 5 inches, a width of 4 inches, and a height of 3 inches. What is the volume of … - Equation of a circle in the xy-plane
In the xy-plane, a circle has center (2, −5) and radius 3. Which equation represents this circle? - Relating area and perimeter of a square
A square has an area of 36 square centimeters. What is the perimeter, in centimeters, of the square? - Angles on a straight line
Two straight lines intersect at a point. One of the four angles formed measures 35°. What is the measure of an angle tha… - Recognising a right triangle from its side lengths
Which of the following sets of side lengths could be the side lengths of a right triangle? - Complementary angles
Two angles are complementary. One of them measures 37°. What is the measure of the other? - Supplementary angles in a given ratio
Two angles are supplementary, and one of them measures 4 times the other. What is the measure, in degrees, of the smalle… - Area of a parallelogram
A parallelogram has a base of 9 centimetres and a height of 4 centimetres. What is its area, in square centimetres? - Splitting a right angle
A ray drawn from the vertex of a right angle divides it into two angles. One of the two angles measures 28°. What is the… - Counting unit cubes in a box
A box has inside dimensions of 2 centimetres by 3 centimetres by 7 centimetres. What is the greatest number of 1-centime… - Three angles on a straight line
Three angles together form a straight angle. Two of them measure 45° and 60°. What is the measure, in degrees, of the th… - Circumference from diameter
A circle has a diameter of 14. What is its circumference? - Acute angles of a right triangle
In a right triangle, one of the acute angles measures 35°. What is the measure, in degrees, of the other acute angle? - Perimeter of an equilateral triangle
An equilateral triangle has sides of length 7. What is its perimeter? - Supplement of an angle
An angle measures 112°. What is the measure, in degrees, of its supplement? - Angle left inside a right angle
A right angle is divided into three angles by two rays from its vertex. Two of the three angles each measure 35°. What i… - Perimeter of a regular pentagon
A regular pentagon has sides of length 6. What is its perimeter? - Fourth angle of a quadrilateral
Three of the interior angles of a quadrilateral measure 80°, 95° and 110°. What is the measure, in degrees, of the fourt… - Area of a rectangular rug
A rectangular rug measures 7 feet by 9 feet. What is its area, in square feet? - Volume of a cube
A cube has edges of length 4 centimetres. What is its volume, in cubic centimetres? - Third angle of an isosceles triangle from its base angles
An isosceles triangle has two angles that each measure 70°. What is the measure, in degrees, of the third angle? - Side of an equilateral triangle from its perimeter
An equilateral triangle has a perimeter of 27 centimetres. What is the length, in centimetres, of each side? - Angle in a right triangle from a labelled vertex
In triangle PQR, angle Q is a right angle and angle P measures 58°. What is the measure, in degrees, of angle R? - Fourth angle of a quadrilateral with two right angles
A quadrilateral has two right angles, and a third angle measures 120°. What is the measure, in degrees, of the fourth an… - Area of a triangular sail
A triangular sail has a base of 4 metres and a height of 6 metres. What is its area, in square metres? - Other angle of a linear pair
Two angles form a linear pair, and one of them measures 65°. What is the measure, in degrees, of the other angle? - Area of an enlarged photograph
A rectangular photograph measures 4 inches by 6 inches. It is enlarged so that each side is doubled. What is the area of… - Diameter from a radius
A circle has a radius of 4.5 centimetres. What is the length, in centimetres, of its diameter? - Vertex angle of an isosceles triangle
In an isosceles triangle, the two equal base angles each measure 70°. What is the measure of the third angle? - Is it a right triangle
A triangle has sides of length 7, 24, and 25. Which of the following statements is true? - Vertical angles at a crossing
Two straight roads cross. One of the four angles formed at the crossing measures 35°. What are the measures of the other… - Area of a rectangle
A rectangular rug is 7 metres long and 3 metres wide. What is its area, in square metres? - Perimeter of a square
Each side of a square tile is 6 centimetres long. What is the perimeter of the tile, in centimetres? - Volume of a rectangular box
A box is 2 centimetres wide, 3 centimetres long, and 4 centimetres tall. What is its volume, in cubic centimetres? - Area of a triangle
A triangle has a base of 8 centimetres and a height of 5 centimetres. What is its area, in square centimetres? - Third angle of a triangle
Two angles of a triangle measure 60° and 70°. What is the measure, in degrees, of the third angle? - Hypotenuse of a 3-4-5 triangle
A right triangle has legs of length 3 and 4. What is the length of its hypotenuse? - Remaining angle in a right triangle named by vertices
In triangle ABC, angle C is a right angle and angle A measures 35°. What is the measure, in degrees, of angle B? - Sine as a ratio of sides
A right triangle has legs of length 3 and 4 and a hypotenuse of length 5. What is the sine of the angle opposite the sid… - Length of a rectangle from its diagonal and width
A rectangle has a width of 5 centimetres and a diagonal of length 13 centimetres. What is the length of the rectangle, i… - Tangent of 45 degrees
In a right triangle, one acute angle measures 45°. What is the tangent of that angle? - Scaling a right triangle
A right triangle has legs of length 6 and 8. A similar right triangle has legs twice as long. What is the length of the … - Diameter from a radius
A circle has a radius of 4 centimetres. What is its diameter, in centimetres? - Degrees in a full turn
How many degrees are there in a full turn around the centre of a circle? - Fraction of a circle in a quarter turn
A sector of a circle has a central angle of 90°. What fraction of the whole circle's area does the sector cover? - Hypotenuse of a 6-8-10 triangle
A ladder's foot is 6 metres from a wall, and its top rests 8 metres up the wall. How long is the ladder, in metres? - Cosine as a ratio of sides
A right triangle has legs of length 6 and 8 and a hypotenuse of length 10. What is the cosine of the angle between the l… - Angles of an isosceles right triangle
A right triangle has two sides of equal length. What is the measure, in degrees, of each of its two acute angles? - Scaling a 5-12-13 triangle
A right triangle has sides of length 5, 12, and 13. A similar triangle has its shortest side of length 15. What is the l… - Value of the sine of 30 degrees
Which of the following is equal to sin 30°? - Diagonal of a rectangular field
A rectangular field is 30 metres wide and 40 metres long. How far is it, in metres, from one corner of the field to the … - Tangent as a ratio of legs
A right triangle has legs of length 6 and 8. What is the tangent of the angle opposite the leg of length 6? - Acute angle of a right triangle from its complement
In a right triangle, one acute angle is 20° more than the other. What is the measure, in degrees, of the larger acute an… - Formula for circumference
A circle has a radius of 5 centimetres. Which expression gives its circumference, in centimetres? - Formula for area
A circle has a radius of 3 centimetres. Which expression gives its area, in square centimetres? - Radius from a diameter
A round coaster has a diameter of 10 centimetres. What is its radius, in centimetres? - Fraction of a circle in a half turn
An arc of a circle has a central angle of 180°. What fraction of the circle's circumference is the arc? - Centre from the equation of a circle
In the xy-plane, which of the following is the centre of the circle x² + y² = 16? - Diameter of a circle from its equation
In the xy-plane, the circle x² + y² = 16 has a radius of 4. What is the length of its diameter? - Central angle of an equal slice
A pizza is cut into 8 equal slices from the centre. What is the central angle, in degrees, of each slice? - Point on a circle given its centre and radius
In the xy-plane, a circle has centre (2, 3) and radius 1. Which of the following points lies on the circle? - Radius in different units
A circle has a diameter of 8 centimetres. What is its radius, in millimetres? - Degrees turned by a clock hand
The minute hand of a clock moves from the 12 to the 3. Through how many degrees does it turn? - Diameter from a compass setting
A circle is drawn with a pair of compasses set to a distance of 6 centimetres between the point and the pencil. What is … - Area of a rectangular rug
A rectangular rug is 5 feet wide and 8 feet long. How many square feet of floor does it cover? - Volume of a small cube
A cube-shaped box has edges of length 2 centimetres. What is the volume of the box, in cubic centimetres? - Area of a triangular sail
A triangular sail has a base of 10 metres and a height of 6 metres. What is the area of the sail, in square metres? - Angles on a straight line
A ray divides a straight angle into two angles. One of the angles measures 110°. What is the measure, in degrees, of the… - Vertically opposite angles
Two straight lines cross. One of the four angles formed measures 35°. What is the measure, in degrees, of the angle dire… - Angle of an equilateral triangle
Each side of a triangle has the same length. What is the measure, in degrees, of each of its angles? - Ladder length from the Pythagorean theorem
The foot of a ladder is 6 feet from the base of a wall, and the top of the ladder rests 8 feet up the wall. How long, in… - Sides of a 45-45-90 triangle
A triangle has angles measuring 90°, 45°, and 45°. Which of the following statements about its sides must be true? - Height of a kite from its string
A kite is flying on a straight string 13 metres long. The end of the string on the ground is 5 metres horizontally from … - Diameter from radius
A circular pond has a radius of 7 metres. What is its diameter, in metres? - Circumference as a multiple of pi
A circle has a radius of 3 inches. Its circumference is kπ inches. What is the value of k? - Area from a diameter as a multiple of pi
A circular table top has a diameter of 10 inches. Its area is kπ square inches. What is the value of k? - Tiles to cover a floor
A floor is 4 metres by 6 metres. How many square tiles, each 1 metre by 1 metre, are needed to cover it exactly? - Volume of a small box
A rectangular box measures 2 centimetres by 3 centimetres by 4 centimetres. What is its volume, in cubic centimetres? - Area of a right triangle from its legs
A right triangle has legs of length 12 and 5. What is its area? - Perimeter of a rectangle
A rectangle is 7 units long and 3 units wide. What is its perimeter, in units? - Complementary angles
Two angles are complementary, and one of them measures 35°. What is the measure, in degrees, of the other angle? - Adjacent angle of a parallelogram
One angle of a parallelogram measures 110°. What is the measure, in degrees, of an angle next to it? - Angle sum of a quadrilateral
The four interior angles of a quadrilateral add up to how many degrees? - Exterior angle of an equilateral triangle
Each interior angle of an equilateral triangle measures 60°. What is the measure, in degrees, of an exterior angle at on… - Screen size from width and height
A television screen is 16 inches wide and 12 inches tall. Screen sizes are given by the length of the diagonal. What is … - Sine of a ramp's angle
A straight ramp is 10 metres long and its top is 6 metres above its bottom. What is the sine of the angle the ramp makes… - Equal acute angles in a right triangle
In a right triangle, the two acute angles are equal. What is the measure, in degrees, of each acute angle? - Radius from a wheel's diameter
A bicycle wheel has a diameter of 70 centimetres. What is its radius, in centimetres? - Degrees turned by a minute hand
In 15 minutes, through how many degrees does the minute hand of a clock turn? - Radius from an area of 49 pi
A circle has an area of 49π square inches. What is its radius, in inches? - Diagonal of a rectangle
A rectangle is 6 units wide and 8 units tall. What is the length of a diagonal of the rectangle? - Central angle of a pizza slice
A circular pizza is cut into 8 equal slices, each with its point at the centre. What is the measure, in degrees, of the … - Square with the same area as a rectangle
A rectangle is 9 centimetres long and 4 centimetres wide. A square has the same area as the rectangle. What is the side … - Number of triangles in a strip of stitches
A quilt pattern is made of identical triangles sewn edge to edge in a straight strip. Each triangle has a base of 5 cent… - Definition of the tangent ratio
In a right triangle, which ratio gives the tangent of one of the acute angles? - Radius from a segment through the centre
Points A and B lie on a circle with centre O, and segment AB passes through O. If AB = 14, what is the length of OA? - Area of a doormat
A rectangular doormat is 7 decimetres long and 3 decimetres wide. What is its area, in square decimetres? - Corresponding angle with parallel lines
Two parallel lines are crossed by a third line. One of the angles formed at the first parallel line measures 130°. What … - Length of a diagonal brace
A rectangular gate is 12 feet wide and 5 feet tall. A straight brace runs from one corner to the opposite corner. How lo… - Distance across a round table
A round table has a radius of 5 feet. What is the distance, in feet, straight across the table through its centre? - Volume of a cube-shaped box
A box is a cube with edges of 6 centimetres. What is its volume, in cubic centimetres? - Fraction of a straight angle
A straight angle measures 180°. What fraction of a straight angle is an angle of 45°? - Height a ladder reaches
A ladder 10 feet long leans against a wall with its foot 6 feet from the base of the wall. How high up the wall, in feet… - Diameter of a coin
A circular coin has a radius of 3.5 millimetres. What is its diameter, in millimetres? - Area of a rectangular rug
A rectangular rug is 8 feet long and 5 feet wide. What is its area, in square feet? - Supplement of an angle
Two angles form a straight line. One measures 100°. What is the measure, in degrees, of the other? - Length of a slanted rope
A rope runs from the top of an 8-metre pole to a peg 15 metres from the base of the pole. How long is the rope, in metre… - Radius from a diameter
A circular pond has a diameter of 9 metres. What is its radius, in metres? - Fence around a vegetable patch
A rectangular vegetable patch is 7 metres long and 3 metres wide. How many metres of fence are needed to go all the way … - Angles of an equilateral triangle
All three sides of a triangle are equal. What is the measure, in degrees, of each of its angles? - Diagonal of a unit square
A square has sides of length 1. The length of its diagonal can be written as √k. What is the value of k? - Diameter of a clock face
A circular clock face has a radius of 6 inches. What is its diameter, in inches? - Edge of a cube from its volume
A cube has a volume of 27 cubic centimetres. What is the length, in centimetres, of one edge of the cube? - Vertical angles with an expression
Angles A and B are vertical angles. If angle A measures 3x degrees and angle B measures 75 degrees, what is the value of… - Diagonal of a rectangle
A rectangle has sides of length 8 and 15. What is the length of its diagonal? - Radius from the centre and a point
A circle in the xy-plane has its centre at the origin and passes through the point (0, 6). What is the radius of the cir… - Area of a rectangular rug
A rectangular rug is 6 feet long and 4 feet wide. What is the area of the rug, in square feet? - Angles that could form a triangle
Which of the following could be the measures of the three angles of a triangle? - Sine of an angle in a right triangle
In right triangle ABC, the right angle is at C. If BC = 6 and AB = 10, what is sin A? - Type of triangle from two angles
A triangle has two angles that measure 30° and 60°. Which of the following describes the triangle? - Equation of a circle centred at the origin
Which of the following is an equation of the circle in the xy-plane with centre (0, 0) and radius 3? - Distance around a field
A rectangular field is 120 metres long and 80 metres wide. A runner runs once around the edge of the field. How far does… - Side lengths that form a right triangle
Which of the following sets of side lengths could form a right triangle? - Tiles to cover a floor
A square tile has sides of 30 centimetres. How many tiles are needed to cover a floor that is 3 metres by 3 metres? - Largest angle of an acute triangle
Which of the following could be the measure of the largest angle of an acute triangle? - Smaller acute angle from a ratio
The two acute angles of a right triangle are in the ratio 1 to 2. What is the measure, in degrees, of the smaller acute … - Distance from the centre to the circle
A circle has a diameter of 14. What is the distance from the centre of the circle to any point on the circle? - Area of a square from its perimeter
A square has a perimeter of 20. What is its area? - Angle of an equilateral triangle
What is the measure of each angle of an equilateral triangle? - Definition of sine in a right triangle
In a right triangle, the sine of an acute angle is the ratio of which two sides? - Centre of a circle from its equation
In the xy-plane, what is the centre of the circle (x − 3)² + (y + 2)² = 9? - Radius from a circumference
A circle's circumference is 10π. What is the radius of the circle?
Easy–Medium (138)
- Circles: area and circumference
A circle has a radius of 6. What is its area? (Use A = πr².) - Pythagorean theorem
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse? - Circumference of a circle
A circle has a radius of 5. What is its circumference? (Use C = 2πr.) - Midpoint in the coordinate plane
What is the midpoint of the segment joining the points (2, 3) and (8, 11)? - Exterior angle theorem / angles in a triangle
What is the measure, in degrees, of angle B? - Area of a trapezoid
What is the area, in square centimeters, of the trapezoid? - Right triangle trigonometry (cosine ratio)
What is the value of cos A? - Base angles of an isosceles triangle
In an isosceles triangle, the angle between the two equal sides measures 40°. What is the measure of each of the other t… - Circumference of a circle
What is the circumference of a circle with radius 7? - Perimeter of a rectangle
A rectangle has length 12 and width 5. What is its perimeter? - Pythagorean theorem: missing leg
A right triangle has a hypotenuse of length 13 and one leg of length 5. What is the length of the other leg? - Sine as a ratio of sides
In right triangle ABC, the right angle is at C. The side opposite angle A has length 3, and the hypotenuse has length 5.… - Interior angles of a polygon
What is the sum of the measures of the interior angles of a pentagon? - Area from a diameter
What is the area of a circle with diameter 10? - Area of a square from its perimeter
A square has a perimeter of 36. What is the area of the square? - Same-side interior angles with a transversal
Lines l and m are parallel and are cut by a transversal. Two same-side interior angles formed measure (3x + 10)° and (2x… - Ladder against a wall
A ladder 10 feet long leans against a vertical wall, with its foot on level ground 6 feet from the base of the wall. How… - Vertical angles with expressions
Two straight lines intersect at a point. A pair of vertical angles formed by the lines measure (2x + 10)° and (3x − 20)°… - Arc length from circumference
A circle has a circumference of 24. An arc of the circle is intercepted by a central angle of 60°. What is the length of… - Same-side interior angle from one angle
Two parallel lines are cut by a transversal. One of the angles formed measures 118°. What is the measure, in degrees, of… - Area of a rectangle from its perimeter
A rectangle has a perimeter of 30 centimetres and a length of 9 centimetres. What is its area, in square centimetres? - Diagonal of a rectangular field
A rectangular field is 30 metres wide and 40 metres long. What is the length, in metres, of a straight path from one cor… - Sides of a regular polygon from its exterior angle
The exterior angles of any polygon, one at each vertex, add to 360°. Each exterior angle of a certain regular polygon me… - Diameter from area
A circle has an area of 49π. What is the length of its diameter? - Length of a kite string
A kite is flying 24 metres above a point on the ground that is 7 metres horizontally from the person holding the string.… - Angles of a triangle in a ratio
The measures of the three angles of a triangle are in the ratio 3 : 4 : 5. What is the measure, in degrees, of the small… - Height of a box from its volume
A rectangular box has a volume of 240 cubic centimetres, a length of 8 centimetres and a width of 5 centimetres. What is… - Radius from an equation with a non-square constant
Which of the following is the radius of the circle in the xy-plane whose equation is x² + y² = 50? - Side from a sine and the hypotenuse
In right triangle ABC, angle C is the right angle, sin A = 3/5, and the hypotenuse AB has length 15. What is the length … - Angles around a point
Three angles meet at a point and together fill the full turn around it. Two of the angles measure 120° and 130°. What is… - Equal sides of an isosceles triangle from its perimeter
An isosceles triangle has a perimeter of 32 centimetres and a base of 12 centimetres. What is the length, in centimetres… - Fraction of a circle in a sector
A sector of a circle has a central angle of 72°. The area of the sector is what fraction of the area of the whole circle… - Cosine from two legs
A right triangle has legs of length 7 and 24. What is the cosine of the angle between the hypotenuse and the leg of leng… - Area of a triangle on a grid
In the xy-plane, a triangle has vertices at (0, 0), (6, 0) and (2, 5). What is the area of the triangle? - Area of a semicircle
A semicircle has a radius of 4. What is its area? - Straight-line distance after two legs
A hiker walks 8 kilometres due east and then 6 kilometres due north. How far, in kilometres, is the hiker from the start… - Angles on a line in a ratio
Three angles that together form a straight angle measure x°, 2x° and 3x°. What is the measure, in degrees, of the larges… - Radius from the distance around a pond
The distance around the edge of a circular pond is 10π metres. What is the radius of the pond, in metres? - Side of a rectangle from its diagonal
A rectangle has a diagonal of length 26 centimetres and one side of length 24 centimetres. What is the length, in centim… - Exact value of sin 60°
What is the value of sin 60°? - Vertical angle with one expression
Two straight lines intersect. One of the two vertical angles formed measures (3x)° and the other measures 75°. What is t… - Central angle for an eighth of a circle
An arc is one-eighth of the way around a circle. What is the measure, in degrees, of the central angle that intercepts t… - Diagonal brace of a rectangular lid
A rectangular lid measures 5 decimetres by 12 decimetres. A straight brace runs from one corner of the lid to the opposi… - Adjacent angles on a line in a ratio
Two adjacent angles together form a straight angle, and one of them measures four times the other. What is the measure, … - Area of a house-shaped composite
A figure consists of a rectangle 8 units wide and 5 units tall with a triangle of height 3 sitting on top of it, the tri… - Guy wire on a mast
A mast stands vertically, and a straight guy wire runs from its top to a point on the level ground 9 metres from its bas… - Value of x from three angle expressions
The three angles of a triangle measure 2x°, 3x° and 4x°. What is the value of x? - Height of a triangle from its area
A triangle has an area of 30 square centimetres and a base of 12 centimetres. What is the height of the triangle, in cen… - Tangent of a ramp's angle
A ramp rises 5 metres over a horizontal run of 12 metres. What is the tangent of the angle the ramp makes with the groun… - Acute angles of a right triangle as expressions
In a right triangle, the two acute angles measure (3x)° and (2x)°. What is the value of x? - Height of a parallelogram from its area
A parallelogram has an area of 48 square centimetres and a base of 8 centimetres. What is its height, in centimetres? - Sine from two legs
A right triangle has legs of length 8 and 15. What is the sine of the angle opposite the leg of length 8? - Angle sum of an octagon
What is the sum, in degrees, of the interior angles of an octagon? - Area of a square frame
A square picture frame has an outer side of 12 inches and a uniform border 2 inches wide. What is the area, in square in… - Circumference in terms of π
A circle has a radius of 7 inches. What is its circumference, in inches? - Perimeter of a right triangle from its legs
The two legs of a right triangle measure 9 centimetres and 12 centimetres. What is the perimeter of the triangle, in cen… - Largest angle of a quadrilateral from a ratio
The measures of the four interior angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. What is the measure, in degr… - Volume of a cylinder in terms of π
A cylindrical water tank has a radius of 3 metres and a height of 10 metres. What is its volume, in cubic metres? - Cosine from a leg and hypotenuse
In a right triangle, the leg adjacent to angle A has length 6 and the hypotenuse has length 10. What is the value of cos… - Base angle from a vertex angle
A roof truss forms an isosceles triangle whose vertex angle (between the two equal sides) measures 48°. What is the meas… - Height of a cone from its volume
A cone has a volume of 12π cubic units and a base radius of 3 units. What is its height, in units? - Number of turns of a wheel
A wheel has a radius of 0.5 metres. How many complete turns does it make while rolling 31.4 metres along the ground? (Us… - Paint needed for a box's outside
A closed box is 7 cm by 4 cm by 3 cm. Which expression gives the total area, in square centimetres, to be painted if all… - Comparing circumferences
Circle P has a radius of 3 and circle Q has a radius of 6. Which statement is true? - How high a ladder reaches
A 15-foot ladder leans against a wall with its foot 9 feet from the base of the wall. How many feet up the wall does the… - Turns of a bicycle wheel over a distance
A bicycle wheel has a radius of 0.35 metres. About how many complete turns does it make in travelling 220 metres? (Use π… - Diagonal of a rectangular field
A rectangular field is 9 metres by 12 metres. A path runs diagonally from one corner to the opposite corner. How long is… - Distance rolled by a wheel
A cart wheel has a radius of 1.5 metres. How many metres does the cart travel while the wheel makes 10 complete turns? (… - Central angle of one slice
A circular cake is cut into 8 equal slices from the centre. What is the measure, in degrees, of the central angle of eac… - Length of a guy rope
A tent pole stands 10 metres tall, and a guy rope runs from its top to a peg 24 metres from its base on level ground. Ho… - Volume of a prism from a triangular cross-section
A wedge-shaped doorstop has a right-triangular cross-section with legs of 4 cm and 5 cm, and it is 7 cm long. What is it… - Total edge length of a rectangular box
A rectangular box has dimensions 4 centimetres by 5 centimetres by 6 centimetres. What is the total length, in centimetr… - Shorter leg of a 30° right triangle
A right triangle has a hypotenuse of length 10 and an acute angle of 30°. What is the length of the leg opposite the 30°… - Acute angles of a right triangle in a given ratio
The measures of the two acute angles of a right triangle are x° and 2x°. What is the value of x? - Fraction of a circle taken by a sector
A sector of a circle has a central angle of 72°. What fraction of the circle's area is the area of the sector? - Area of a ring between concentric circles
Two circles share the same centre and have radii 3 and 5. The area of the region between the two circles can be written … - Inscribed angle from its intercepted arc
In a circle, an inscribed angle intercepts an arc of measure 100°. What is the measure of the inscribed angle, in degree… - Volume of a rectangular pool
A rectangular swimming pool is 10 metres long, 4 metres wide, and 2 metres deep throughout. How many cubic metres of wat… - Obtuse angle at a parallel line
Lines m and n are parallel, and a third line crosses both of them. One of the angles formed at line m measures 65°. What… - Area from a circumference
A circle has a circumference of 10π. Its area can be written as kπ. What is the value of k? - Capacity of a fish tank
A fish tank is 40 centimetres long, 25 centimetres wide, and 30 centimetres tall. How many cubic centimetres of water do… - Height of a triangle from its area
A triangle has an area of 24 square units and a base of 8 units. What is the height of the triangle, in units? - Area of a parallelogram
A parallelogram has a base of 10 units and a height of 4 units. What is its area, in square units? - Rectangle against a triangle on the same base
A rectangle and a triangle both have a base of 6 units and a height of 4 units. How much larger, in square units, is the… - Water in a rectangular pool
A rectangular pool holds water to a depth of 2 metres everywhere. The surface of the water is 25 metres by 10 metres. Ho… - Length of a rectangle with a square's area
A square has an area of 36 square centimetres. A rectangle with the same area has a width of 4 centimetres. What is the … - Perimeter from an area and a length
A rectangle has an area of 54 square units and a length of 9 units. What is its perimeter, in units? - Classifying a triangle from two angles
Two angles of a triangle measure 40° and 75°. Which of the following describes the triangle? - Middle angle on a straight line
Three angles on a straight line measure 50°, x°, and 70°. What is the value of x? - Base angles from a vertex angle
An isosceles triangle has a vertex angle of 100°. What is the measure, in degrees, of each base angle? - Same-side interior angles at parallel lines
Two parallel lines are cut by a transversal. A pair of same-side interior angles measures 3x° and x°. What is the value … - Curved edge of a semicircle
A semicircle has a diameter of 12. The length of its curved edge can be written as kπ. What is the value of k? - Diameter of a circle through a given point
In the xy-plane, a circle has its centre at the origin and passes through the point (0, 5). What is the length of its di… - Area of a quarter circle
A quarter circle has a radius of 4. Its area can be written as kπ. What is the value of k? - Length of one sixth of a circle
An arc makes up one sixth of a circle with radius 12. The length of the arc can be written as kπ. What is the value of k… - Cosine from side lengths
In a right triangle, the leg adjacent to angle A has length 8, the leg opposite angle A has length 15, and the hypotenus… - Tiles needed for a floor
A rectangular floor measures 5 metres by 4 metres. It is to be covered completely with square tiles that are 50 centimet… - Water in a pool filled part way
A swimming pool is 25 metres long and 10 metres wide, and its floor is flat. How many cubic metres of water are in the p… - Value of x from an angle bisector
Ray OC bisects angle AOB. If angle AOC measures (4x − 5)° and angle COB measures (2x + 25)°, what is the value of x? - Acute angles of a right triangle in a ratio
In a right triangle, the two acute angles are in the ratio 2 : 1. What is the measure, in degrees, of the smaller acute … - Length of a zip line
A zip line runs in a straight line from the top of a vertical 30-metre tower to a point on level ground 40 metres from t… - Opposite leg from a tangent ratio
In a right triangle, tan A = 3/4, and the leg adjacent to angle A has length 8. What is the length of the leg opposite a… - Radius from the centre and a point on the circle
In the xy-plane, a circle has its centre at (3, −2), and the point (3, 4) lies on the circle. What is the radius of the … - Lowest point on a circle from its equation
In the xy-plane, a circle has the equation (x − 1)² + (y + 4)² = 36. What is the y-coordinate of the lowest point on the… - Small cubes packed in a larger cube
A cube-shaped box has inside edges of 20 centimetres. It is packed completely full with solid cubes that have edges of 5… - Area of a border around a photo
A rectangular photograph 12 centimetres by 8 centimetres is mounted in the centre of a rectangular card, leaving a borde… - Angle between two equal sides
In triangle XYZ, XY = XZ, and angle Y measures 50°. What is the measure, in degrees, of angle X? - Width of a screen from its diagonal
A rectangular television screen has a diagonal of 17 inches and a height of 8 inches. What is the width of the screen, i… - Diameter from a point and the centre
In the xy-plane, the point (7, 1) lies on a circle whose centre is (−3, 1). What is the diameter of the circle? - Bags of soil for a raised bed
A raised garden bed is 3 metres long, 4 metres wide, and is to be filled with soil to a depth of half a metre. Each bag … - Area of a rectangle when its width is tripled
A rectangle is 2 metres wide and 7 metres long. If its width is tripled and its length is unchanged, what is the new are… - Largest angle of an isosceles triangle
In an isosceles triangle, the two equal angles each measure 35°. What is the measure, in degrees, of the third angle? - Value of x from a pair of vertical angles
Two straight lines cross. One of the pairs of vertical angles they form measures (3x + 10)° and (5x − 30)°. What is the … - Depth reached by a diagonal pipe
A straight pipe 15 metres long runs from ground level down into a trench, reaching a point 9 metres horizontally from wh… - Radius from a circle's equation
In the xy-plane, a circle has the equation (x + 2)² + (y − 7)² = 25. What is the radius of the circle? - Highest point on a circle
In the xy-plane, a circle has centre (2, 7) and radius 3. What is the y-coordinate of the highest point on the circle? - Floor area minus a hearth
A room's floor is a rectangle 12 feet by 9 feet, and a rectangular stone hearth 2 feet by 3 feet is set into it. How man… - Area of a triangular sail
A triangular sail has a base of 10 feet and a height of 6 feet. What is its area, in square feet? - Third angle of a right triangle from one acute angle
A right triangle has one angle of 27°. What is the measure, in degrees, of its other acute angle? - Perimeter of an enlarged triangle
A triangle with sides 6, 8, and 10 is enlarged by a scale factor of 5. What is the perimeter of the enlarged triangle? - Distance across a pond
A surveyor stands at a point on the bank of a pond. A pole on the opposite bank is 20 metres away in a straight line, an… - Sine from the sides of a right triangle
In a right triangle, the leg opposite angle B has length 12 and the hypotenuse has length 13. What is sin B? - Area of a circular pond with an approximate pi
A circular pond has a radius of 2 metres. Using 3.14 for π, what is its area, in square metres? - Distance from the centre to a point on the circle
In the xy-plane, a circle has centre (0, 0) and passes through the point (0, −4). What is the diameter of the circle? - Missing angle of a quadrilateral
The angles of a quadrilateral measure 80°, 95°, 110°, and x°. What is the value of x? - Shorter leg from a ratio and the hypotenuse
The legs of a right triangle are in the ratio 3 to 4, and the hypotenuse has length 20. What is the length of the shorte… - Diameter from a circle's equation
In the xy-plane, a circle has the equation (x − 2)² + (y + 5)² = 16. What is the length of the circle's diameter? - Perimeter of a rectangle from its area and length
A rectangle has an area of 36 square units and a length of 9 units. What is the perimeter of the rectangle? - Area of a square with a rectangle's perimeter
A square has the same perimeter as a rectangle that is 3 units by 9 units. What is the area of the square? - Cosine from a sine
In a right triangle, one acute angle θ has sin θ = 0.6. What is the value of cos θ? - Height of a triangle from its area
A triangle has a base of 10 and an area of 35. What is its height? - Value of x from two angle expressions
Two angles of a triangle measure 3x° and 2x°, and the third angle measures 80°. What is the value of x? - Diagonal of a square
A square has sides of length 6. What is the length of its diagonal, rounded to the nearest tenth? - Diameter from a circumference
The circumference of a circle is 12π. What is the diameter of the circle? - Area of a parallelogram
A parallelogram has a base of 9 and a height of 4. What is its area? - Sum of the other three angles at an intersection
Two lines intersect, forming four angles. One of the angles measures 40°. What is the sum, in degrees, of the other thre… - Diameter from an area
A circle has an area of 64π square units. What is the length of its diameter? - Cosine from two legs
In right triangle PQR, the right angle is at Q, PQ = 9, and QR = 12. What is the value of cos P?
Medium (137)
- Special right triangles (45-45-90)
In a 45°-45°-90° right triangle, each leg has length 5. What is the length of the hypotenuse? - Volume of a cylinder
A cylinder has a radius of 3 and a height of 10. What is its volume? (Use V = πr²h.) - Trigonometric ratios (SOH-CAH-TOA)
In a right triangle, the side opposite angle θ has length 3 and the hypotenuse has length 5. What is sin θ? - Similar triangles
Two triangles are similar with a ratio of 2 to 3. A side of the smaller triangle measures 8. What is the length of the c… - Distance in the coordinate plane
What is the distance between the points (1, 2) and (4, 6) in the coordinate plane? - Area of a sector
A circle has a radius of 6. What is the area of a sector with a central angle of 60°? - Volume of a sphere
A sphere has a radius of 3. What is its volume? (Use V = (4/3)πr³.) - Slope in the coordinate plane
What is the slope of the line passing through the points (-2, 1) and (4, 10)? - Trigonometric ratios (tangent)
In a right triangle, the side opposite angle θ has length 8 and the side adjacent to angle θ has length 15. What is the … - Corresponding sides of similar triangles
Triangle ABC is similar to triangle DEF, with vertices A, B, C corresponding to D, E, F respectively. AB = 6, DE = 9, an… - Centre of a circle from its equation
A circle in the xy-plane has the equation (x − 2)² + (y + 3)² = 25. What are the coordinates of its centre? - Volume of a cylinder
A cylinder has a base of radius 3 and a height of 10. What is its volume? - 45-45-90 triangle
An isosceles right triangle has legs of length 7. What is the length of its hypotenuse? - 30-60-90 triangle
In a right triangle, one acute angle measures 30° and the side opposite it has length 4. What is the length of the longe… - Perimeter of a similar triangle
A triangle has side lengths 3, 4 and 5. A second triangle is similar to it, and the shortest side of the second triangle… - Radius from the equation of a circle
In the xy-plane, the graph of x² + y² = 36 is a circle. What is the radius of the circle? - Surface area of a rectangular box
A closed rectangular box has dimensions 4 by 5 by 6. What is its total surface area? - Vertex angle of an isosceles triangle
In triangle PQR, PQ = PR and the measure of angle Q is 65°. What is the measure of angle P? - Area of a border around a rectangle
A rectangular garden measures 10 metres by 6 metres. A path of uniform width 1 metre runs around the outside of the gard… - Area from circumference
A circle has a circumference of 16π. What is the area of the circle? - Each angle of a regular hexagon
Each interior angle of a regular hexagon measures how many degrees? - Area between a square and its inscribed circle
A circle is inscribed in a square with side length 10, touching all four sides. What is the area of the region inside th… - Tangent meets radius at a right angle
A line is tangent to a circle of radius 5 at point T. Point P lies on the tangent line, 12 units from T. What is the dis… - Longer leg from the hypotenuse of a 30-60-90 triangle
The hypotenuse of a 30°-60°-90° right triangle has length 10. What is the length of the longer leg? - Exterior angle at the base of an isosceles triangle
In triangle ABC, AB = AC and angle A measures 100°. Side CB is extended past B to point D. What is the measure, in degre… - Equal volumes, unknown height
A cube with edges of length 6 has the same volume as a rectangular box with a base measuring 4 by 9. What is the height … - Sector area from the circle's area
A circle has an area of 36π. What is the area of a sector of the circle with a central angle of 90°? - Cosine of a ramp's angle
A straight ramp is 13 feet long, and its top is 5 feet higher than its bottom. What is the cosine of the angle the ramp … - Alternate interior angles with expressions
Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (4x − 10)° and (2x + 30)°. What… - Area of a square from its diagonal
A square has a diagonal of length 10. What is the area of the square? - Volume of a triangular prism
A prism has a base that is a right triangle with legs of length 3 and 4, and the prism has a length of 10. What is its v… - Angle where sine equals cosine
For an acute angle θ, sin θ = cos θ. What is the measure of θ, in degrees? - Longest side opposite the largest angle
In triangle ABC, angles A and B each measure 50°. Which side of the triangle is the longest? - Area of a rhombus from its diagonals
A rhombus has diagonals of length 15 and 16. What is the area of the rhombus? (The diagonals of a rhombus are perpendicu… - Surface area of a cube from its volume
A cube has a volume of 27 cubic centimetres. What is its total surface area, in square centimetres? - Rectangle area from a perimeter and a ratio
A rectangle's length is twice its width, and its perimeter is 36. What is the area of the rectangle? - Volume ratio from halving the radius
Cylinder A has a radius of 4 and a height of 5. Cylinder B has a radius of 2 and the same height. The volume of cylinder… - Circumference from a point on the circle
In the xy-plane, a circle is centred at the origin and passes through the point (6, 8). What is the circumference of the… - Sine of a ladder's angle
A 5-metre ladder leans against a wall, and its top touches the wall 4 metres above the ground. What is the sine of the a… - Exterior angle equals the two remote interior angles
In triangle ABC, angle A measures 40°, and the exterior angle at vertex C measures 115°. What is the measure, in degrees… - Corresponding angles with expressions
In a figure, lines p and q are parallel and are crossed by a third line. Angle 1 and angle 5 are corresponding angles. I… - Change in a cube's surface area
The edge of a cube is increased from 2 centimetres to 3 centimetres. By how many square centimetres does the cube's tota… - Radius from the centre and a point on the circle
In the xy-plane, a circle has centre (2, −1) and passes through the point (5, 3). What is the radius of the circle? - Leg of an isosceles right triangle from its hypotenuse
An isosceles right triangle has a hypotenuse of length 10. What is the length of one of its legs? - Other angles of an isosceles triangle with an obtuse angle
One angle of an isosceles triangle measures 100°. What is the measure, in degrees, of each of the other two angles? - Square with a quarter circle removed
A quarter circle of radius 6 is cut from one corner of a square with side length 6, using that corner as the centre of t… - Shadow at a 45° sun angle
A vertical flagpole is 12 metres tall. When the sun's rays make a 45° angle with the level ground, how long, in metres, … - Third exterior angle of a triangle
The exterior angles of a triangle, one at each vertex, add up to 360°. Two of a triangle's exterior angles measure 110° … - Volume of a square pyramid
A pyramid has a square base with sides of length 6 and a height of 5. What is its volume? (The volume of a pyramid is on… - Packing 2-centimetre cubes
A crate measures 6 cm by 8 cm by 10 cm on the inside. Cubes with edges of 2 cm are packed into the crate with no gaps. H… - Missing coordinate of a point on a circle
The point (x, 4) lies on the circle x² + y² = 25 in the xy-plane, and x > 0. What is the value of x? - Area of a square after one side is shortened
A square has sides of length 10. One pair of opposite sides is shortened to 7.5 while the other pair stays at 10, making… - Arc length as a fraction of the circumference
A circle has a radius of 9. What is the length of an arc that is intercepted by a central angle of 120°, expressed as a … - Exact value of cos 45°
What is the value of cos 45°? - Base angle of an isosceles triangle from its apex
In triangle ABC, AB = BC. If angle B measures 36°, what is the measure, in degrees, of angle A? - Volume of a cylindrical tank from its width
A cylindrical water tank is 10 metres across and 3 metres deep. What is its volume, in cubic metres? - Area of a path around a circular bed
A circular flower bed has an area of 49π square feet. A path of uniform width 1 foot runs around the outside of the bed.… - Area of one slice of a pizza
A circular pizza with a diameter of 16 inches is cut into 8 equal slices. What is the area of one slice, in square inche… - Sum of two special-angle values
What is the value of sin 30° + cos 60°? - Volume after tripling one dimension
A rectangular box measures 3 by 4 by 5 units. If its height of 5 units is tripled while the other dimensions stay the sa… - Another point on the same circle
In the xy-plane, a circle is centred at the origin and passes through the point (5, 12). Which of the following points a… - Height from a shadow ratio
At the same moment, a vertical post 4 feet tall casts a shadow 6 feet long, and a nearby vertical flagpole casts a shado… - Surface area of a box
A closed rectangular box has dimensions 6 cm by 4 cm by 3 cm. What is its total surface area, in square centimetres? - Ladder length from its foot distance and angle
A ladder leans against a vertical wall, making a 60° angle with the level ground. The foot of the ladder is 2 metres fro… - Interior angle of a regular pentagon
A garden bed is shaped like a regular pentagon. What is the measure of each of its interior angles? - Volume of a prism with a triangular base
A tent is shaped like a triangular prism. Its triangular end has a base of 4 feet and a height of 3 feet, and the tent i… - Area of a sector
A circle has an area of 36π square inches. What is the area, in square inches, of a sector with a central angle of 30°? … - Distance from a building using a 45° angle
From a point on level ground, the angle of elevation to the top of a 30-metre tower is 45°. How many metres is the point… - Angle relationships with a transversal
Two parallel roads are crossed by a straight path. At the first road, the acute angle between the path and the road meas… - Area of a trapezoid-shaped field
A field is shaped like a trapezoid with parallel sides of 8 metres and 14 metres and a perpendicular distance of 5 metre… - Sector area as a fraction of the circle
A circular pizza with an area of 36π square inches is cut into equal slices, each with a central angle of 120°. What is … - Height of a tree from an angle of elevation
From a point 10 metres from the base of a tree, on level ground, the angle of elevation to the top of the tree is 60°. W… - Length in a triangle cut by a parallel line
In triangle ABC, point D is on side AB and point E is on side AC such that DE is parallel to BC. AD = 4, DB = 2, and BC … - Volume of water in a half-full trough
A trough has the shape of a rectangular box 40 cm long, 25 cm wide, and 20 cm deep. It is filled with water to half its … - Arc length of a quarter circle
A circle has a radius of 5 inches. What is the length, in inches, of an arc that measures 90°? (Give your answer as a mu… - Trig ratio of a 45° angle
In an isosceles right triangle, what is the value of tan of either acute angle? - Arc length from a central angle and circumference
A circular running track has a circumference of 240 metres. A runner goes from one marker to the next, and the two marke… - Area of a quarter circle
A quarter-circle garden bed has a radius of 4 metres. What is its area, in square metres? - Height reached on a wall from a 60° angle and base distance
A ladder makes a 60° angle with the level ground, and its foot is 5 metres from the wall. How high up the wall, in metre… - Vertical drop from slope length and angle
A straight ramp is 18 metres long and makes a 30° angle with the horizontal ground. What is the vertical height of the t… - Apex angle from a base angle
In an isosceles triangle, each of the two equal angles measures 52°. What is the measure, in degrees, of the third angle… - Volume of a box built from a square base
A square-based box has a base with a perimeter of 28 centimetres and a height of 2 centimetres. What is its volume, in c… - Arc length from a central angle
A circular garden path has a total length (circumference) of 24 metres. A section of the path subtends a central angle o… - Area of a sector in terms of π
A circle has a radius of 6 centimetres. What is the area, in square centimetres, of a sector with a central angle of 120… - Horizontal distance from a slope length and angle
A slide is 16 metres long and makes a 60° angle with the ground. How many metres of ground does the slide cover horizont… - Perimeter of a 30-60-90 triangle from its hypotenuse
A right triangle has a hypotenuse of length 10 and one acute angle of 30°. What is the perimeter of the triangle? Round … - Triangle angle from an exterior angle
In triangle PQR, the exterior angle at P measures 110° and angle Q measures 45°. What is the measure, in degrees, of ang… - Triangle angles from described relationships
In triangle ABC, the measure of angle A is twice the measure of angle B, and the measure of angle C is 20° more than the… - Adjacent leg from a sine value
In a right triangle, one acute angle θ satisfies sin θ = 0.28, and the hypotenuse has length 25. What is the length of t… - Leg from a tangent ratio and the hypotenuse
In a right triangle, tan θ = 3/4 for one of the acute angles θ, and the hypotenuse has length 25. What is the length of … - Similar right triangles: scaling a leg
A right triangle has legs of length 6 and 8. A second, similar right triangle has a hypotenuse of length 30. What is the… - Distance from an external point to a circle's centre via a tangent
A line through point P is tangent to a circle of radius 5 at point T, and PT = 12. What is the distance from P to the ce… - Diameter length from the endpoints of a diameter
In the xy-plane, the points (−1, 2) and (7, 8) are the endpoints of a diameter of a circle. What is the length of the di… - Arc length as a fraction of the circumference
A circle has a circumference of 16 units. What is the length of an arc of the circle whose central angle measures 135°? - Largest angle from angles in terms of k
The three angles of a triangle measure 2k°, 3k°, and 4k°. What is the measure, in degrees, of the largest angle of the t… - Fourth angle of a quadrilateral
Three angles of a quadrilateral measure 90°, 110°, and 75°. What is the measure, in degrees, of the fourth angle? - Longest side of a similar triangle
Two triangles are similar. The sides of the smaller triangle are 3, 4, and 6. The shortest side of the larger triangle i… - Half-chord from a circle's equation
In the xy-plane, a circle has the equation x² + y² = 25. The horizontal line y = −3 meets the circle at two points. What… - Volume of a tank with an approximate pi
A cylindrical water tank has a radius of 3 metres and a height of 5 metres. Using 3.14 for π, what is the volume of the … - Bags of compost for a trapezium-shaped bed
A garden bed is shaped like a trapezium whose parallel sides are 6 metres and 10 metres long and are 4 metres apart. One… - Consecutive angles of a parallelogram
In parallelogram ABCD, angle A measures 70°. What is the measure, in degrees, of angle B? - Distance to a cliff from an angle of elevation
From a boat, the angle of elevation to the top of a vertical cliff 60 metres high is 45°. How far, in metres, is the boa… - Radius from the centre and a point by distance
In the xy-plane, a circle has its centre at (2, 3) and passes through the point (5, 7). What is the radius of the circle… - Quarter of a circle's circumference
A circle has a radius of 8. The length of an arc cut off by a central angle of 90° can be written as kπ. What is the val… - Ice cubes from a block
A block of ice is a cube with edges of 10 centimetres. It is cut into smaller cubes with edges of 5 centimetres. What is… - Interior angle of a regular hexagon
What is the measure, in degrees, of each interior angle of a regular hexagon? - Exact value of tan 60°
What is the value of tan 60°? - Angle in a triangle from a linear pair
In triangle DEF, side DE is extended beyond E to a point G, and angle FEG measures 110°. If angle D measures 40°, what i… - Point on a circle from its x-coordinate
In the xy-plane, the circle x² + y² = 50 passes through a point with x-coordinate 5 and positive y-coordinate. What is t… - Area of a trapezium-shaped sign
A road sign is a trapezium with parallel sides of 5 and 11 decimetres, which are 4 decimetres apart. What is the area of… - Surface area of a box
A closed rectangular box measures 3 by 4 by 5 centimetres. What is its total surface area, in square centimetres? - Scaling a photograph
A photograph 4 inches wide and 10 inches tall is enlarged so that its width becomes 6 inches, keeping the same proportio… - Height from a 45° angle of elevation
From a point 12 metres from the base of a wall, the angle of elevation to the top of the wall is 45°. How tall is the wa… - Centre from the equation of a circle
In the xy-plane, a circle has the equation (x − 5)² + (y + 1)² = 9. What is the x-coordinate of the rightmost point on t… - Distance rolled by a wheel in one turn
A wheel has a radius of 0.5 metres. How far, in metres, does the wheel travel along the ground in one complete turn? Use… - Volume difference between two cubes
One cube has edges of 5 centimetres and another has edges of 3 centimetres. By how many cubic centimetres does the large… - Height of a lamp post from a model
A scale model of a lamp post is 5 centimetres tall and casts a 2-centimetre shadow on the model's base. The real lamp po… - Height of a kite from its string angle
A kite string 18 metres long makes an angle of 30° with the level ground. How many metres above the ground is the kite? - Equation of a circle from centre and radius
In the xy-plane, a circle has centre (3, −4) and radius 10. Which equation represents the circle? - Angle that is twice another in a triangle
In triangle ABC, the measure of angle A is twice the measure of angle B, and angle C measures 60°. What is the measure, … - Side from a sine and the hypotenuse
In right triangle ABC, the right angle is at C and sin A = 5/13. If AB = 26, what is the length of BC? - Surface area of a box
A closed rectangular box has length 5, width 3 and height 2. What is the total surface area of the box? - Radius from an arc length and its central angle
In a circle, a central angle of 45° cuts off an arc of length 3π. What is the radius of the circle? - Rise in water depth from an added volume
A rectangular tank has a base measuring 50 centimetres by 20 centimetres and contains water to a depth of 30 centimetres… - Exterior angle of a quadrilateral
In quadrilateral PQRS, angles P, Q, and R measure 70°, 100°, and 110°. What is the measure, in degrees, of the exterior … - Tangent from a hypotenuse and one leg
A right triangle has a hypotenuse of length 26 and one leg of length 24. What is the tangent of the angle opposite the o… - Perimeter of a semicircle
A semicircle has a diameter of 10. What is the perimeter of the semicircle, including its straight edge? - Volume ratio of two cubes
A cube has edges of length 4. A second cube has edges twice as long. The volume of the second cube is how many times the… - Length of a midsegment
In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. If BC = 14, what is the length of MN? - Angle between a tangent and a radius
Line l is tangent to a circle with centre O at the point P. What is the measure, in degrees, of the angle between line l… - Height of a cylinder from its volume
A cylinder has a volume of 100π cubic units and a radius of 5 units. What is its height? - Angle of a ramp from its length and rise
A straight ramp is 10 metres long and rises 4 metres from one end to the other. What is the angle, in degrees, that the … - Side length from a ratio of areas
Two similar triangles have areas of 9 and 36. A side of the smaller triangle has length 4. What is the length of the cor… - Hypotenuse of a 30-60-90 triangle
In a right triangle with acute angles of 30° and 60°, the shorter leg has length 5. What is the length of the hypotenuse… - Ratio of the areas of two circles
Two circles have radii of 3 and 4. What is the ratio of the area of the smaller circle to the area of the larger circle? - Angles in a 2:3:4 ratio
The measures of a triangle's three angles are in the ratio 2:3:4. What is the measure, in degrees, of the largest angle? - Perimeter of a right triangle from its legs
The legs of a right triangle measure 9 and 12. What is the perimeter of the triangle?
Hard (138)
- Right triangles and the Pythagorean theorem
In a right triangle, one leg is 5 and the hypotenuse is 13. What is the length of the other leg? - Arc length
A circle has a radius of 4. What is the length of an arc that subtends a central angle of 90°? - Equation of a circle
A circle is defined by the equation (x − 3)² + (y + 2)² = 25. What is the radius of the circle? - Converting degrees to radians
What is 135° expressed in radians? - Circles in the coordinate plane: completing the square and tangent lines
What is the length of segment PT? - Arc length and sector area with radian measure
What is the measure, in degrees, of the central angle of this sector? - Right triangle trigonometry with similarity scaling
What is the area, in square units, of triangle ABC? - Exterior angle of a triangle
One exterior angle of a triangle measures 130°. One of the two interior angles not adjacent to it measures 45°. What is … - Arc length from a central angle
A circle has radius 6. What is the length of the arc intercepted by a central angle of 60°? - Effect of scaling on volume
The length of each edge of a cube is doubled. The volume of the new cube is how many times the volume of the original cu… - Using tangent to find a side
In a right triangle, one acute angle θ satisfies tan θ = 3/4. The leg adjacent to θ has length 12. What is the length of… - Sine and cosine of complementary angles
If sin 35° = cos x°, and 0 < x < 90, what is the value of x? - Distance between two points
In the xy-plane, what is the distance between the points (1, 2) and (7, 10)? - Triangle angles in a ratio
The measures of the three angles of a triangle are x°, 2x° and 3x°. What is the measure, in degrees, of the largest angl… - Central angle from an arc length
A circle has radius 9. An arc of the circle has length 3π. What is the measure, in degrees, of the central angle that in… - Area of a trapezoid
A trapezoid has parallel sides of lengths 8 and 12, and the perpendicular distance between them is 5. What is the area o… - Triangle inequality at the boundary
Two sides of a triangle have lengths 5 and 12. Which of the following could NOT be the length of the third side? - Volume ratio when dimensions change
A cylinder has radius r and height h. A second cylinder has radius 2r and height h/2. The volume of the second cylinder … - Volume of a cube from its surface area
A cube has a total surface area of 150 square inches. What is the volume of the cube, in cubic inches? - Angle formed by an angle bisector
In triangle ABC, angle A measures 50° and angle B measures 60°. The bisector of angle C meets side AB at point D. What i… - Cylinder volume from its base circumference
The base of a right circular cylinder has a circumference of 12π, and the cylinder has a height of 5. What is the volume… - Tangent from sine
For an acute angle θ, sin θ = 5/13. What is the value of tan θ? - Least possible perimeter with an integer side
Two sides of a triangle have lengths 4 and 9. If the third side has an integer length, what is the least possible perime… - Volume of a hemisphere
A solid hemisphere has a radius of 3. What is its volume? (The volume of a sphere of radius r is (4/3)πr³.) - Central angle when a chord equals the radius
In a circle with centre O, points A and B lie on the circle and the chord AB has the same length as the radius. What is … - Heights from shadows
At the same moment, a person 1.8 metres tall casts a shadow 2.4 metres long, and a nearby tree casts a shadow 12 metres … - Volume by displacement
A rectangular tank has a base measuring 50 centimetres by 40 centimetres and contains water. When a stone is fully subme… - Testing points against a circle's equation
In the xy-plane, a circle has the equation (x + 3)² + (y − 1)² = 16. Which of the following points lies on the circle? - Midsegment of a triangle
In triangle ABC, point D is the midpoint of side AB and point E is the midpoint of side AC. If DE = 7, what is the lengt… - Curved edge of a pizza slice
A circular pizza has a radius of 12 inches. A slice is cut with a central angle of 150°. What is the length, in inches, … - Hypotenuse of an isosceles right triangle from its area
An isosceles right triangle has an area of 18. What is the length of its hypotenuse? - Tangent from cosine
For an acute angle θ, cos θ = 0.8. What is the value of tan θ? - Similar triangles from their perimeters
Two triangles are similar. The perimeter of the smaller is 18 and the perimeter of the larger is 27. A side of the small… - Circle through the origin with centre on the y-axis
A circle in the xy-plane has its centre at (0, 4) and passes through the origin. Which equation represents the circle? - Height of an equilateral triangle
An equilateral triangle has sides of length 8. What is the height of the triangle? - Number of sides from the angle sum
The interior angles of a polygon add up to 1,080°. How many sides does the polygon have? - Radius of a cone from its volume
A cone has a volume of 12π cubic centimetres and a height of 4 centimetres. What is the radius of its base, in centimetr… - Scaling a triangle by its longest side
A triangle has sides of length 6, 8 and 10. A similar triangle has a longest side of length 25. What is the length of th… - Equal volumes, different radii
Cylinder A has a radius of 2 and a height of 9. Cylinder B has a radius of 3 and the same volume as cylinder A. What is … - Central angle from a sector's area
A sector of a circle with radius 6 has an area of 12π. What is the measure, in degrees, of the sector's central angle? - Perimeter of a scaled right triangle
A right triangle has sides of length 5, 12 and 13. A similar triangle has a hypotenuse of length 39. What is the perimet… - Area between two concentric circles
Two circles share the same centre. One has a radius of 3 and the other a radius of 5. What is the area of the ring-shape… - Hypotenuse from the longer leg of a 30-60-90 triangle
In a 30°-60°-90° right triangle, the longer leg has length 6√3. What is the length of the hypotenuse? - Unknown side in similar triangles with a decimal answer
Triangles ABC and DEF are similar, with A, B, C corresponding to D, E, F. AB = 6, DE = 15 and DF = 9. What is the length… - Label area on a can
A cylindrical can has a radius of 5 centimetres and a height of 10 centimetres. A paper label covers the curved side of … - Whether a point is inside, on, or outside a circle
In the xy-plane, a circle has centre (1, 2) and radius 3. Which of the following points lies OUTSIDE the circle? - Largest angle of a 7-24-25 triangle
A triangle has sides of length 7, 24 and 25. What is the measure, in degrees, of its largest angle? - Length of a chord from two points on a circle
The points (6, 8) and (−6, 8) both lie on the circle x² + y² = 100 in the xy-plane. What is the length of the chord join… - Angle in a right triangle from its sides
A right triangle has a hypotenuse of length 2√2 and one leg of length 2. What is the measure, in degrees, of the angle o… - Side ratio from an area ratio
Two similar triangles have areas of 18 and 50 square units. What is the ratio of a side of the smaller triangle to the c… - Cube around a sphere
A sphere of radius 3 fits exactly inside a cube, touching all six faces. What is the volume of the cube? - Tangent of the smaller acute angle
The legs of a right triangle are 6 and 8. What is the tangent of the smaller of the triangle's two acute angles? - Largest angle from three expressions
The angles of a triangle measure (2x + 10)°, (3x − 10)°, and x°. What is the measure, in degrees, of the largest angle? - Volume of a pyramid with a square base
A pyramid has a square base with side length 6 units and a height of 5 units. What is its volume, in cubic units? - Arc length from a central angle
A circle has a circumference of 16 centimetres. What is the length, in centimetres, of an arc that subtends a central an… - Perimeter of a similar triangle
Triangle ABC has sides of length 5, 6, and 10. Triangle DEF is similar to triangle ABC, and its longest side has length … - Side of a square from its area, then perimeter
A square rug has an area of 36 square feet. A border strip is to run around all four edges of the rug. How many feet of … - Radius from a chord and its distance from the centre
A chord of a circle has length 24 and is 5 units from the centre of the circle. What is the radius of the circle? - Corresponding side from a perimeter ratio
Two similar triangles have perimeters of 7 centimetres and 21 centimetres. The shortest side of the smaller triangle is … - Volume of a hollow box's walls
A closed wooden box has outer dimensions 10 cm by 10 cm by 10 cm and walls 1 cm thick on every side. What is the volume … - Which point is inside a given circle
A circle in the xy-plane has centre (1, −2) and radius 5. Which of the following points lies INSIDE the circle? - Value of x from three angle expressions
The three angles of a triangle measure (4x + 2)°, (3x + 10)°, and (5x + 24)°. What is the value of x? - Comparing a cube and a pyramid
A cube has edge length 3 units. A square pyramid has the same base as one face of the cube and the same height as the cu… - Length of a zip wire from its drop and angle
A straight zip wire runs from a platform to the ground, making a 30° angle with the level ground. The platform is 12 met… - Equation of a circle from its centre and a point
In the xy-plane, a circle has centre (2, 3) and passes through the point (5, 7). Which equation represents the circle? - Central angle from an arc fraction
A circle has circumference 18 centimetres. An arc of the circle has length 6 centimetres. What is the measure, in degree… - Trig ratio for the larger acute angle
A right triangle has legs of length 7 and 24. What is the sine of the larger acute angle? - Leg of an isosceles right triangle from its hypotenuse
A square napkin is folded along a diagonal to make a right triangle whose longest side is 12 centimetres. What is the le… - Radius from centre and a point on the circle
In the xy-plane, a circle has centre (3, 4) and passes through (7, 1). What is the radius of the circle? - Diameter from a circle's equation
In the xy-plane, a circle has the equation (x − 1)² + (y + 2)² = 49. What is the length of a diameter of the circle? - Tangent of the smaller acute angle
A right triangle has legs of length 5 and 12. What is the tangent of the smaller acute angle? - Side from a cosine value and the hypotenuse
In triangle ABC, angle C is a right angle, cos A = 0.8, and AB = 15. What is the length of BC? - Centre of a circle from its general-form equation
In the xy-plane, the equation x² + y² − 6x + 4y − 12 = 0 represents a circle. What is the sum of the x- and y-coordinate… - Central angle from an inscribed angle in the same circle
Points A, B, and C lie on a circle with centre O. The inscribed angle ABC measures 45° and intercepts arc AC. What is th… - Volume of a sphere as a multiple of pi
The volume of a sphere with radius r is (4/3)πr³. The volume of a sphere with radius 3 can be written as kπ. What is the… - Volume of a cube from its surface area
A cube has a total surface area of 54 square centimetres. What is its volume, in cubic centimetres? - Volume of a cylindrical tank as a multiple of pi
A cylindrical water tank has a radius of 3 metres and a height of 10 metres. Its volume can be written as kπ cubic metre… - Surface area of a rectangular box
A closed rectangular box measures 4 cm by 5 cm by 6 cm. What is its total surface area, in square centimetres? - Area of a trapezoid
A trapezoid has parallel sides of length 8 and 12 and a height of 5. What is its area? - Largest angle of a quadrilateral
The four angles of a quadrilateral measure x°, 2x°, 3x°, and 4x°. What is the measure, in degrees, of the largest angle? - Longest side of a similar triangle
Two triangles are similar. The smaller has sides of length 6, 8, and 10. The shortest side of the larger triangle has le… - Sine from a tangent
In a right triangle, tan A = 5/12, where A is one of the acute angles. What is the value of sin A? - Height reached by a ladder at 60 degrees
A 10-metre ladder leans against a wall, making a 60° angle with the ground. The height it reaches up the wall can be wri… - Shorter leg of a 30-60-90 triangle
In a right triangle, one acute angle measures 30° and the hypotenuse has length 8. What is the length of the shorter leg… - Area of a can's label
A cylindrical can has a radius of 4 centimetres and a height of 10 centimetres. A label wraps once around the curved sur… - Total edge length of a cube from its volume
A cube has a volume of 64 cubic centimetres. What is the total length, in centimetres, of all 12 of its edges? - Base angles of an isosceles triangle
Two sides of a triangle have equal length, and the angle between them measures 36°. What is the measure, in degrees, of … - Remote interior angle from an exterior angle
The exterior angle at one vertex of a triangle measures 110°, and one of the two remote interior angles measures 45°. Wh… - Altitude to the hypotenuse
A right triangle has legs of length 9 and 12. What is the length of the altitude drawn to the hypotenuse? - Adjacent leg from a cosine
In a right triangle, cos A = 0.6, where A is an acute angle, and the hypotenuse has length 20. What is the length of the… - Sum of a circle's centre coordinates
In the xy-plane, a circle has the equation (x − 3)² + (y + 2)² = 25. What is the sum of the coordinates of the circle's … - Area of a semicircle from a circumference
A circle has a circumference of 12π. The area of a semicircle of this circle can be written as kπ. What is the value of … - Volume of a cone as a multiple of pi
A cone has a base radius of 3 and a height of 4. Its volume can be written as kπ. What is the value of k? - Surface area of a square-based box
A closed box has a square base with side 5 and a height of 2. What is its total surface area? - Angle from an exterior angle of a triangle
In triangle PQR, angle P measures 40°. Side QR is extended beyond R to a point S, and angle PRS measures 110°. What is t… - Altitude to the hypotenuse
A right triangle has legs of length 5 and 12 and a hypotenuse of length 13. What is the length of the altitude drawn fro… - Height of a pole from an angle of elevation
From a point 12 metres from the base of a vertical pole, the angle of elevation to the top of the pole is 30°. What is t… - Complementary angle identity
In right triangle DEF, angle F is the right angle and the measure of angle D is 50°. Which of the following is equal to … - Cones filled from a cylindrical can
A cylindrical can with radius 5 and height 12 is full of water. The water is poured into empty cone-shaped cups, each wi… - Volume of a triangular prism
A right prism has a base that is a right triangle with legs of length 6 and 8, and the prism has a height of 10. What is… - Side of a similar triangle from an area ratio
Triangle ABC is similar to triangle DEF, and the area of triangle ABC is 9 times the area of triangle DEF. If AB = 12, w… - Fifth angle of a pentagon
The interior angles of a pentagon have a sum of 540°. Four of the interior angles of a certain pentagon each measure 100… - Area of an isosceles right triangle from its hypotenuse
An isosceles right triangle has a hypotenuse of length 8. What is the area of the triangle? - Distance from a circle's centre to a chord
A circle with centre O has radius 10. Chord AB of the circle has length 12. What is the distance from O to chord AB? - Angle formed by a bisector in an isosceles triangle
In triangle ABC, AB = AC and angle A measures 36°. The bisector of angle B meets side AC at point D. What is the measure… - Volume of a pyramid on a square base
A pyramid has a square base with sides of 6 centimetres and a height of 8 centimetres. What is the volume of the pyramid… - Height of a tree from a stick's shadow
At the same moment, a vertical stick 6 feet tall casts a shadow 9 feet long, and a nearby tree casts a shadow 15 feet lo… - Height gained on a ramp
A straight ramp is 50 metres long and makes an angle of 30° with the horizontal ground. How many metres higher is the to… - Rise of a road from its angle
A straight section of road 40 metres long rises at an angle of 30° to the horizontal. By how many metres does the road r… - Comparing two cylinders
Two cylinders have the same height. The radius of the larger one is twice the radius of the smaller one. The volume of t… - Area left after cutting corners from a square
A square sheet of card has sides of 10 centimetres. A 2-centimetre square is cut from each of its four corners. What is … - Angle from a relation between angles
In triangle GHK, angle G is 20° less than twice angle H, and angle K measures 20°. What is the measure, in degrees, of a… - Longest side from a ratio and a perimeter
The sides of a triangle are in the ratio 3 : 4 : 5, and its perimeter is 60 centimetres. What is the length, in centimet… - Horizontal distance from a slant length
A 20-metre cable runs from the top of a mast to the ground, making an angle of 60° with the ground. How far, in metres, … - Length of a gentle ramp
A wheelchair ramp rises 1 metre and makes an angle of 5° with the ground. To the nearest metre, how long is the ramp? - Arc length as a coefficient of pi
A circle has a radius of 6. An arc of the circle is cut off by a central angle of 150°. The length of the arc can be wri… - Number of small boxes that fit in a crate
A crate is 6 feet by 4 feet by 2 feet on the inside. How many boxes that are 2 feet by 2 feet by 1 foot can be packed in… - Volume of half a cylinder
A trough is shaped like half of a cylinder, cut along its length, with radius 3 feet and length 8 feet. Its volume can b… - Exterior angle of a twelve-sided regular polygon
A regular polygon has 12 sides. What is the measure, in degrees, of one of its exterior angles? - Corresponding side of a similar triangle
Triangle ABC is similar to triangle DEF, with AB corresponding to DE. If AB = 6, DE = 9, and BC = 4, what is the length … - Length of a slide from its angle
A playground slide makes a 60° angle with the ground, and its bottom end is 15 feet horizontally from the point below it… - Sector area as a coefficient of pi
A circle has radius 8. A sector with a central angle of 45° has an area that can be written as kπ. What is the value of … - Point with equal coordinates on a circle
In the xy-plane, the circle x² + y² = 50 passes through the point (a, a), where a > 0. What is the value of a? - Exterior angle from two remote interior angles
In triangle ABC, angle A measures 40° and angle B measures 75°. Side BC is extended beyond C to a point D. What is the m… - Tangent from a cosine
In right triangle PQR, the right angle is at R, and cos P = 0.8. What is the value of tan P? - Volume of a triangular prism
A prism has a triangular base with a base length of 8 and a height of 5, and the prism is 12 units long. What is the vol… - Volume of a cube from its surface area
A cube has a total surface area of 150 square units. What is its volume, in cubic units? - Hypotenuse from a tangent and a leg
In right triangle ABC, the right angle is at C and tan A = 2. If BC = 8, what is the length of AB, rounded to the neares… - Chord under a 60-degree central angle
Points A and B lie on a circle with centre O, and angle AOB measures 60°. If the radius of the circle is 6, what is the … - Area of a path around a pond
A circular pond with a radius of 5 metres is surrounded by a path 1 metre wide. What is the area of the path? - Angle between a perpendicular and a parallel line
Lines l and m are parallel. A third line crosses both, making an acute angle of 35° with line l. A fourth line, perpendi… - Distance of a boat from its start
A boat sails 20 kilometres due east and then 21 kilometres due north. How far, in kilometres, is the boat from its start… - Longest diagonal of a box
A rectangular box measures 4 by 6 by 10. What is the length of the longest diagonal inside the box, rounded to the neare… - Possible length of a triangle's third side
Triangle PQR has PQ = 5 and QR = 7. Which of the following could be the length of PR? - Sum of the x-intercepts of a circle
In the xy-plane, the circle (x − 1)² + (y − 1)² = 25 crosses the x-axis at two points. What is the sum of the x-coordina… - Perimeter from a tangent and the hypotenuse
In right triangle ABC, the right angle is at C, tan A = 3/4, and AB = 25. What is the perimeter of the triangle? - Equation of a circle tangent to both axes
In the xy-plane, a circle with radius 3 lies in the first quadrant and is tangent to both the x-axis and the y-axis. Whi… - Radius by completing two squares
In the xy-plane, the equation x² + y² + 8x − 2y + 8 = 0 describes a circle. What is the radius of the circle?
Very hard (136)
- Angles formed by parallel lines
Two parallel lines are cut by a transversal. One of the eight angles measures 47°. Which of the following is NOT a possi… - Complementary angle trig identity (sin/cos)
In a right triangle, angles A and B are the two acute angles, so A + B = 90°. If sin A = 0.6, what is the value of cos B… - Area of a regular polygon (hexagon)
A regular hexagon has a side length of 4. What is its area? (A regular hexagon of side s has area (3√3/2)s².) - Volume of a cone
A cone has a radius of 6 and a height of 10. What is its volume? (Use V = (1/3)πr²h.) - Circles in the coordinate plane: chord length from a line–circle intersection
In the xy-plane, the graph of x² + y² − 12x − 4y + 15 = 0 is a circle. The line 3x − 4y = 25 intersects this circle at t… - Similar triangles formed by the altitude to the hypotenuse of a right triangle
What is the perimeter of triangle ABC? - Volumes of similar solids (partially filled cone)
What is the volume, in cubic inches, of the water in the cup? - Tangent lines, central angles, and arc length
What is the length of minor arc AB? - Triangle inequality
Two sides of a triangle have lengths 7 and 10. If the third side has an integer length, what is the greatest possible le… - Area of a sector
A circle has radius 9. What is the area of the sector formed by a central angle of 40°? - Trigonometric ratios of the two acute angles
Triangle ABC is a right triangle with its right angle at C. If sin A = 5/13, what is the value of cos B? - Pythagorean identity
For an acute angle θ, sin θ = 0.6. What is the value of cos θ? - Equation of a circle from centre and radius
Which equation represents the circle in the xy-plane with centre (−1, 4) and radius 3? - Cone and cylinder with the same base and height
A cone and a cylinder have the same base radius, 3, and the same height, 4. How many times must the cone be filled with … - Similar triangles from a parallel segment
In triangle ABC, point D lies on side AB and point E lies on side AC such that segment DE is parallel to side BC. If AD … - Diameter from a circle's general-form equation
The equation x² + y² − 6x + 8y = 0 defines a circle in the xy-plane. What is the length of the circle's diameter? - Sine from tangent
In right triangle ABC, angle C is the right angle and tan A = 2. What is the value of sin A? - Area ratio of similar triangles
Two triangles are similar, and the ratio of their corresponding side lengths is 2 : 5. The area of the smaller triangle … - Total surface area of a closed cylinder
A closed right circular cylinder has a radius of 3 and a height of 7. What is its total surface area? - Circle from the endpoints of a diameter
In the xy-plane, the points (−2, 1) and (4, 9) are the endpoints of a diameter of a circle. Which equation represents th… - Volume remaining after a hole is drilled
A cube has edges of length 4. A cylindrical hole of radius 1 is drilled straight through the cube from the centre of one… - Complementary angles inside sine and cosine
For an acute angle with measure x°, sin(x°) = cos((2x − 15)°). What is the value of x? - Chord length from its distance to the centre
A circle has a radius of 5. A chord of the circle is 3 units from the centre. What is the length of the chord? - Area of a right triangle from its hypotenuse and an angle
A right triangle has a hypotenuse of length 10 and one acute angle measuring 60°. What is the area of the triangle? - Area of a stadium shape
A running track's infield consists of a rectangle 100 metres long and 60 metres wide with a semicircle attached to each … - Circle matched to a square's perimeter
A circle's circumference is equal to the perimeter of a square with side length π. What is the area of the circle? - Surface area after cutting a cube
A cube with edges of length 3 is cut into 27 cubes with edges of length 1. What is the total surface area of all 27 smal… - Two angles whose sine and cosine agree
In a right triangle, the two acute angles measure a° and b°, and sin a° = cos b° = 1/2. What is the value of b − a? - Centre from the general form of a circle
The equation x² + y² + 4x − 6y − 3 = 0 represents a circle in the xy-plane. What are the coordinates of the centre of th… - Rise of a ramp from its tangent and length
A straight ramp rises at an angle whose tangent is 3/4. The sloped surface of the ramp is 20 metres long. How many metre… - Perimeter of the midpoint triangle
The midpoints of the three sides of a triangle with perimeter 36 are joined to form a smaller triangle. What is the peri… - Surface area of a sphere from its volume
A sphere has a volume of 36π. What is its surface area? (For a sphere of radius r, the volume is (4/3)πr³ and the surfac… - Sum of a sine and a cosine
In right triangle ABC, angle C is the right angle, AC = 8 and BC = 6. What is the value of sin A + cos A? - Surface area of an open-topped cylinder
An open-topped cylindrical can has a radius of 2 and a height of 3. What is the total surface area of the can, including… - Sine of a complementary angle
For an acute angle B, cos B = 5/13. What is the value of sin(90° − B)? - Exterior angle at the vertex of an isosceles triangle
In triangle ABC, AB = AC and angle B measures 70°. Side BA is extended past A to point D. What is the measure, in degree… - Triangle area with the height inside a rectangle
A rectangle is 12 units long and 9 units tall. A triangle is drawn with one side along the rectangle's bottom edge (the … - Circumference of a circle from its equation
The equation (x − 2)² + (y + 5)² = 9 represents a circle in the xy-plane. What is the circumference of the circle? - Height of a kite from an angle of elevation
A kite string is 20 metres long and makes a 30° angle with the level ground. Assuming the string is straight, how high a… - Edge ratio from a volume ratio
The volume of cube A is 8 times the volume of cube B. The length of an edge of cube A is how many times the length of an… - Exterior angle with two expressions
In a triangle, two of the interior angles measure (x + 10)° and (2x − 5)°. The exterior angle at the third vertex measur… - Equation of a circle tangent to the x-axis
In the xy-plane, a circle has centre (2, −3) and is tangent to the x-axis. Which equation represents the circle? - Angle from a tangent of 1/√3
For an acute angle θ, tan θ = 1/√3. What is the measure of θ, in degrees? - Rhombus with a right-angle diagonal relation
A kite-shaped garden has diagonals of length 10 metres and 10 metres that meet at right angles. What is its area, in squ… - Expanded form of a circle equation
Which of the following equations represents the circle with centre (3, −1) and radius 5 in the xy-plane? - Relationship between sine and cosine of complementary angles
In right triangle PQR, angle R is the right angle and sin P = 0.6. What is cos Q? - Angle in a triangle with an altitude
In triangle PQR, angle P measures 40° and angle Q measures 90°. An altitude is drawn from Q to side PR, meeting it at po… - Effect of scaling on volume
A sculptor makes a small clay model of a statue and then a full-size version that is 4 times as tall, with the same shap… - Centre from an expanded equation
A circle in the xy-plane has the equation x² + y² + 8x − 6y = 0. What are the coordinates of its centre? - Trig ratio in a similar triangle
Right triangle JKL has legs JK = 9 and KL = 12, with the right angle at K. Triangle MNP is similar to triangle JKL, with… - Triangle inequality applied
Which of the following could be the lengths of the three sides of a triangle? - Lateral surface area of a cylinder
A cylindrical label wraps exactly once around the side of a can with radius 4 cm and height 10 cm, with no overlap. What… - Leg length from a cosine and the hypotenuse
In a right triangle, the hypotenuse has length 26 and one acute angle θ satisfies cos θ = 5/13. What is the length of th… - Angle in a regular polygon's diagonal
In a regular hexagon ABCDEF, what is the measure of angle ACD, formed by diagonal AC and side CD? - Effect of doubling the radius of a cylinder
A cylindrical tank's radius is doubled while its height stays the same. How does the volume change? - Radius from an equation in general form
The equation x² + y² − 10x + 4y + 13 = 0 represents a circle in the xy-plane. What is the radius of the circle? - Half-chord from radius and distance to centre
A circle has radius 10. A chord of the circle is 6 units from the centre. What is the length of the chord? - Sum of the sine and cosine of an angle
A right triangle has legs of length 7 and 24. Let θ be the acute angle opposite the leg of length 7. What is the value o… - Distance of a chord from the centre
A circle has a radius of 13 units, and one of its chords has length 10 units. How far, in units, is the chord from the c… - Area of a right triangle from a cosine and the hypotenuse
In right triangle ABC, the right angle is at C, the hypotenuse AB has length 20, and cos A = 0.6. What is the area of th… - Angle formed by an angle bisector
In triangle ABC, the measure of angle B is 50° and the measure of angle C is 70°. The bisector of angle A meets side BC … - Co-interior angles on parallel lines
Two parallel lines are cut by a transversal. Two interior angles on the same side of the transversal measure (3x + 10)° … - Interior angles from an exterior angle
In triangle ABC, the exterior angle at vertex C measures 130°, and the measure of angle A is 20° greater than the measur… - Angle split by the altitude of an isosceles triangle
In triangle PQR, PQ = PR and the measure of angle P is 40°. Point S lies on QR so that PS is perpendicular to QR. What i… - Perimeter of a similar triangle from its longest side
A triangle has sides of length 6, 9, and 12. A similar triangle has its longest side of length 20. What is the perimeter… - Number of sides from the interior angle sum
The sum of the interior angles of a polygon is 1,440°. How many sides does the polygon have? - Base side of a box from its volume
A rectangular box has a square base, and its height is twice the side length of the base. The volume of the box is 128 c… - Centre and radius from the general form
In the xy-plane, the circle x² + y² − 6x + 4y − 12 = 0 has centre (h, k) and radius r. What is the value of h + k + r? - Cone volume from a matching cylinder
A cone and a cylinder have the same radius and the same height. The volume of the cylinder is 60 cubic inches. What is t… - Radius of a cylinder as tall as it is wide
A cylinder's height is equal to its diameter, and its volume is 54π cubic centimetres. What is the radius of the cylinde… - Volume factor when a sphere's radius doubles
The radius of a sphere is doubled. The volume of the larger sphere is how many times the volume of the original sphere? - Equal angles in a pentagon
Three angles of a pentagon measure 100°, 110°, and 120°. The other two angles are equal to each other. What is the measu… - Angle formed by an angle bisector
In triangle ABC, angle B measures 50° and angle C measures 70°. Point D lies on side BC so that segment AD bisects angle… - Counting possible integer third sides
Two sides of a triangle have lengths 7 and 12. If the third side has an integer length, how many different lengths are p… - Acute angle from a sine-cosine identity
In a right triangle, A and B are the two acute angles, and the measure of angle A is twice the measure of angle B. What … - Perimeter of a square from its diagonal
The diagonal of a square has length 8√2. What is the perimeter of the square? - Tower height from a tangent
From a point on level ground 50 metres from the base of a tower, the angle of elevation to the top of the tower has a ta… - Constant in a circle's equation from a tangent line
In the xy-plane, a circle with centre (2, −1) is tangent to the line y = 3. The equation of the circle is (x − 2)² + (y … - Central angle from an arc length
An arc of a circle with radius 6 has length 4π. What is the measure, in degrees, of the central angle that subtends the … - Sector area from a central angle
A circle has an area of 36π. A sector of the circle has a central angle of 60°. The area of the sector is kπ. What is th… - Volume factor when a cone changes shape
The radius of a cone is doubled and its height is halved. The new volume is how many times the original volume? - Cube with a cylindrical hole
A cube with edge length 6 has a cylindrical hole of radius 1 drilled straight through from one face to the opposite face… - Area from a hypotenuse and one leg
A right triangle has a hypotenuse of length 25 and one leg of length 7. What is the area of the triangle? - Altitude from the segments of a hypotenuse
In a right triangle, the altitude to the hypotenuse divides the hypotenuse into segments of length 4 and 9. What is the … - Hypotenuse of an isosceles right triangle from its area
An isosceles right triangle has an area of 32. Its hypotenuse can be written as k√2. What is the value of k? - Distance from an angle of depression
From the top of a vertical cliff 30 metres high, the angle of depression to a boat at sea has a tangent of 0.5. How far,… - Chord cut by a vertical line on a circle
In the xy-plane, the circle x² + y² = 25 and the line x = 3 intersect at two points. What is the distance between those … - Distance to the centre along a tangent
A line from a point P touches a circle of radius 6 at point T, so that PT is tangent to the circle. If PT = 8, what is t… - Area of a square outside its inscribed circle
A circle is drawn inside a square of side 10 so that it touches all four sides. The area of the square that lies outside… - Circumference from a quarter-sector's area
A sector of a circle has a central angle of 90° and an area of 9π. The circumference of the whole circle can be written … - Longest side of a similar triangle from its perimeter
A triangle has sides of length 3, 4, and 5. A similar triangle has a perimeter of 60. What is the length of the longest … - Angle from two relationships in a triangle
In triangle ABC, the measure of angle A is twice the measure of angle B, and the measure of angle C equals the sum of th… - Sides of a regular polygon from an exterior angle
Each exterior angle of a regular polygon measures 20°. How many sides does the polygon have? - Equal sides of an isosceles triangle from its height
In triangle ABC, AB = AC. Point D lies on BC so that AD is perpendicular to BC. If BC = 10 and AD = 12, what is the leng… - Volume of a box from its three face areas
A rectangular box has faces with areas 6, 8, and 12 square units (one of each size on opposite pairs of faces). What is … - Empty space around a sphere in a cube
A sphere of radius 3 fits exactly inside a cube, touching all six faces. The volume of the cube not occupied by the sphe… - Percent change in volume from a longer edge
The edge length of a cube is increased by 50%. By what percent does the volume of the cube increase? - Side of a triangle from a parallel segment
In triangle PQR, point S lies on side PQ and point T lies on side PR so that segment ST is parallel to side QR. If PS = … - Cosine of the larger acute angle
In a right triangle, the shorter leg is one third as long as the hypotenuse. What is the cosine of the larger acute angl… - Arc length from a central angle
A circle has radius 9. What is the length of an arc of the circle that is cut off by a central angle of 120°? - Height of water after pouring between containers
A cylindrical jug with an inside radius of 2 centimetres is filled with water to a height of 10 centimetres. All the wat… - Length from a midsegment
In triangle ABC, points M and N are the midpoints of sides AB and AC. Segment MN has length 6, and the perimeter of tria… - Angle from a sine-cosine identity
For acute angles, sin 35° = cos k°. What is the value of k? - Area between a quarter circle and a triangle
A quarter circle has radius 4, with its two straight edges meeting at the centre O. The chord joining the ends of the ar… - Height of water from a melted sphere
A solid sphere of ice with radius 3 centimetres melts completely into an empty cylindrical glass with inside radius 3 ce… - Perimeter of a quarter circle
A quarter circle has radius 10. What is the perimeter of the quarter circle, including its two straight edges? - Perimeter of a similar triangle from one pair of sides
Triangle PQR has sides 4, 5, and 6. Triangle XYZ is similar to PQR, and its shortest side has length 6. What is the peri… - Angle whose tangent is 1
In a right triangle, the two legs have equal length. What is the measure of each acute angle? - Volume of material in a hollow pipe
A pipe is a cylinder 10 centimetres long with an outer radius of 6 centimetres and a circular hole of radius 3 centimetr… - Angle between a bisector and a side
In triangle RST, RS = RT and angle S measures 36°. The bisector of angle R meets ST at U. What is the measure, in degree… - Cosine of an angle from side ratio
In right triangle KLM, the right angle is at M, and the hypotenuse KL is twice as long as leg KM. What is the measure of… - Radius from a sector's area and angle
A sector of a circle has a central angle of 120° and an area of 12π. What is the radius of the circle? - Height of a box with the same volume as a cube
A cube has edges of 4 centimetres. A rectangular box with a square base of side 8 centimetres has the same volume as the… - Curved surface of a cylinder
A cylindrical tin has radius 3 centimetres and height 10 centimetres. The area of its curved surface, not counting the t… - Angle formed by extending a side of an isosceles triangle
In triangle KLM, KL = KM and angle L measures 40°. Side LM is extended beyond M to a point N. What is the measure, in de… - Cosine of the other acute angle
In a right triangle, sin A = 5/13, where A is one of the acute angles. What is the cosine of the other acute angle? - Radius where circumference and area match numerically
For a certain circle, the number of units in its circumference equals the number of square units in its area. What is th… - Length of a tangent from an external point
A circle has centre O and radius 9. A point P lies 15 units from O, and a line from P touches the circle at a single poi… - Angle between two angle bisectors
In triangle PQR, angle P measures 50° and angle Q measures 60°. The bisectors of angles Q and R meet at point I. What is… - Perimeter of a similar right triangle
Two right triangles are similar. The smaller has legs of length 6 and 8, and the larger has a hypotenuse of length 35. W… - Volume left after removing a cone
A cylinder has a base area of 36 square centimetres and a height of 10 centimetres. A cone with the same base and height… - Highest point of a circle tangent to the x-axis
In the xy-plane, a circle with centre (3, 4) is tangent to the x-axis. What is the y-coordinate of the highest point on … - Cone height matching a cylinder's volume
A cone and a cylinder have the same radius. The cylinder's height is 4, and the cone's volume equals the cylinder's volu… - Segment cut by an angle bisector
In triangle ABC, point D lies on side BC so that AD bisects angle BAC. If AB = 8, AC = 12, and BC = 15, what is the leng… - Sine of the complement
Angles A and B are the two acute angles of a right triangle, and sin A = 5/13. What is the value of sin B? - Angle in a triangle with a second isosceles triangle inside
In triangle ABC, AB = AC and angle A measures 36°. Point D lies on AC so that BD = BC. What is the measure, in degrees, … - Diameter of a circle through three points
In the xy-plane, a circle passes through the points (0, 0), (8, 0), and (0, 6). What is the length of the circle's diame… - Surface area of a sphere from its volume
A sphere has a volume of 36π cubic units. What is its surface area? - Altitude from the segments it cuts on the hypotenuse
In right triangle ABC, the right angle is at C. The altitude from C meets the hypotenuse AB at point D, with AD = 4 and … - Surface area of a box from its volume
A rectangular box has a volume of 480 cubic units, a length of 10 units, and a width of 8 units. What is the total surfa… - Distance from a vertex to the centroid
A median of a triangle has length 12. How far is the triangle's centroid from the vertex at which that median starts? - Height of a tower from two angles of elevation
From a point on level ground, the angle of elevation to the top of a tower is 30°. From a point 40 metres closer to the … - Perimeter of a hexagon inscribed in a circle
A regular hexagon is inscribed in a circle of radius 6. What is the perimeter of the hexagon? - Remote interior angles from an exterior angle
In triangle ABC, side BC is extended beyond C, forming an exterior angle of 115 degrees at C. The measure of angle A is … - Sine from a tangent ratio
In a right triangle, tan A = 8/15 for one of the acute angles A. What is the value of sin A? (Enter your answer as a fra… - Tangency to the x-axis fixes the center
In the xy-plane, a circle with center (4, k), where k > 0, has radius 5 and is tangent to the x-axis. What is the value …
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