A circular theater stage has a designated performance section shaped as a circular sector … — ACT

A circular theater stage has a designated performance section shaped as a circular sector with a central angle measuring $\frac{2\pi}{5}$ radians. If the arc length along the outer edge of this sector is $8\pi$ feet, what is the area, in square feet, of the performance section?

A
$20\pi$
B
$40\pi$
$80\pi$ Answer
D
$160\pi$
E
$400\pi$

Answer

The area of the performance sector is

80\pi

square feet.

First, determine the radius of the circle using the radian arc length formula

s = r\theta

. Substituting

s = 8\pi

and

\theta = \frac{2\pi}{5}

yields

r = 20

feet. Next, calculate the area of the circular sector using

A = \frac{1}{2}r^2\theta = \frac{1}{2}(20)^2\left(\frac{2\pi}{5}\right) = 80\pi

square feet (or equivalently

A = \frac{1}{2}rs = \frac{1}{2}(20)(8\pi) = 80\pi

square feet).

Step-by-Step Solution

Use the arc length formula in radians to solve for the radius (

r

Since

s = r\theta

, substituting

s = 8\pi

and

\theta = \frac{2\pi}{5}

gives

8\pi = r \cdot \frac{2\pi}{5}

, so

r = 8\pi \cdot \frac{5}{2\pi} = 20

feet.

The radius of the circular stage is required to calculate the sector area.

Calculate the area of the sector using the formula

A = \frac{1}{2}r^2\theta

(or

A = \frac{1}{2}rs

A = \frac{1}{2}(20)^2\left(\frac{2\pi}{5}\right) = \frac{1}{2}(400)\left(\frac{2\pi}{5}\right) = 200\left(\frac{2\pi}{5}\right) = 80\pi

square feet.

Applying the known radius and central angle into the sector area formula yields the final area.

Key Concept

Calculating sector area from arc length and central angle in radians

Estimated Time:1m 15s

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