A sector of a circle with a radius of 10 units has a total perimeter of 20 + 5\pi units. W… — GRE General Test

A sector of a circle with a radius of $10$ units has a total perimeter of $20 + 5\pi$ units. What is the area of this sector?

The area of the sector is

25\pi

The total perimeter of a sector with radius

r

and arc length

L

is given by

P = 2r + L

. Substituting

r = 10

gives

20 + 5\pi = 20 + L

, so

L = 5\pi

. Using the sector area formula

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

, the area is

\frac{1}{2} \times 10 \times 5\pi = 25\pi

Set up the formula for the perimeter of a sector.

P = 2r + L

, where

r = 10

is the radius and

L

is the arc length.

A sector's perimeter is bounded by two straight radii and one curved arc.

Solve for the arc length

L

20 + 5\pi = 2(10) + L \implies L = 5\pi

Subtracting the combined length of the two radii (

20

) isolates the arc length.

Calculate the area of the sector.

Sector Area =

\frac{1}{2} r L = \frac{1}{2} (10)(5\pi) = 25\pi

The area of a sector can be directly evaluated using half the product of its radius and arc length.

Perimeter, Arc Length, and Area of a Circular Sector

Find the central angle

\theta

first: Since

L = 5\pi

and circumference

C = 2\pi(10) = 20\pi

, the fraction of the circle is

\frac{5\pi}{20\pi} = \frac{1}{4}

, which corresponds to

\theta = 90^\circ

. The area is then

\frac{1}{4} \times \pi(10^2) = 25\pi

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