In a circle with center O, a sector has a central angle measuring \frac{2\pi}{5} radians a… — ACT

In a circle with center $O$ , a sector has a central angle measuring $\frac{2\pi}{5}$ radians and an arc length of $6\pi$ centimeters. What is the area, in square centimeters, of the sector?

A
$15\pi$
B
$\frac{9\pi}{5}$
$45\pi$ Answer
D
$90\pi$
E
$225\pi$

Answer

The area of the sector is

45\pi

square centimeters.

To find the area of the sector, first determine the radius

r

of the circle using the arc length formula for radians,

s = r\theta

. Substituting

s = 6\pi

and

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

yields

6\pi = r \left(\frac{2\pi}{5}\right)

, which simplifies to

r = 15

centimeters. Then, apply the sector area formula

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

. Substituting

r = 15

and

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

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters. Alternatively, using

A = \frac{1}{2} r s

directly gives

A = \frac{1}{2} (15)(6\pi) = 45\pi

square centimeters.

Step-by-Step Solution

Find the radius of the circle using the arc length formula in radians.

r = 15

centimeters

The arc length formula for an angle in radians is

s = r\theta

. Substituting

s = 6\pi

and

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

gives

6\pi = r \left(\frac{2\pi}{5}\right)

, which yields

r = 6\pi \cdot \frac{5}{2\pi} = 15

cm.

Calculate the area of the sector using the sector area formula.

A = 45\pi

square centimeters

The area of a sector with central angle

\theta

in radians is

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

. Substituting

r = 15

and

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

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = \frac{1}{2} (225) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters.

Key Concept

Arc Length and Sector Area in Radians

Estimated Time:1m 0s

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