Geometry and Trigonometry practice question

Answer: 216

Let r be the radius and θ the central angle in radians. Then arc length s = rθ = 6π and sector area A = (1/2)r²θ = 15π. Substituting rθ = 6π into the area equation gives (1/2)r(rθ) = (1/2)r(6π) = 3πr = 15π, so r = 5. Then θ = 6π/5 radians, and converting: (6π/5)(180°/π) = 216°. Check: area = (216/360)·π·5² = 0.6·25π = 15π ✓ and arc = (216/360)·2π·5 = 6π ✓. Common errors: solving for r but then reporting the radian value 6π/5 ≈ 3.77; or dropping the factor of 1/2 in the area formula, which gives r = 2.5 and θ = 2.4π radians, or 432° — an impossible central angle greater than 360°, which is the clue that a step went wrong.

Common wrong answers

If you answered 5

That's the radius r, which is only the first step — now use θ = 6π/r = 6π/5 radians and convert to 216 degrees.

If you answered 432

You dropped the ½ in the sector area formula: solving r·(rθ) = 15π with rθ = 6π gives r = 2.5 and θ = 2.4π, or 432°. A central angle can't exceed 360°, which is the clue. With A = ½r²θ, 3πr = 15π gives r = 5 and θ = 6π/5, or 216°.

If you answered 3.77

You found the angle correctly in radians (6π/5 ≈ 3.77) but the question asks for degrees: multiply by 180/π to get 216.

If you answered 6π/5

That's the angle in radians. Convert: (6π/5)(180°/π) = 216 degrees.

More Geometry and Trigonometry questions

• Hard
  In a right triangle, one leg is 5 and the hypotenuse is 13. What is the length of the other leg?
• Hard
  A circle has a radius of 4. What is the length of an arc that subtends a central angle of 90°?
• Hard
  A circle is defined by the equation (x − 3)² + (y + 2)² = 25. What is the radius of the circle?
• Hard
  What is 135° expressed in radians?
• Hard
  What is the length of segment PT?
• Hard
  What is the area, in square units, of triangle ABC?