In a circle centered at point O, the ratio of the area of sector AOB to the area of the en… — GRE General Test

In a circle centered at point $O$ , the ratio of the area of sector $AOB$ to the area of the entire circle is $3:8$ . If the total perimeter of sector $AOB$ is $24 + 9\pi$ , what is the radius of the circle?

Answer: 12

Answer

The radius of the circle is 12.

Since the ratio of the sector area to the total circle area is

3:8

, the sector accounts for

\frac{3}{8}

of the entire circumference

2\pi r

, making the arc length

L = \frac{3}{8}(2\pi r) = \frac{3}{4}\pi r

. The perimeter of the sector is

2r + L = 2r + \frac{3}{4}\pi r

. Equating this to

24 + 9\pi

gives

2r = 24

, so

r = 12

Step-by-Step Solution

Relate sector area ratio to arc length

Arc length

L = \frac{3}{4}\pi r

The fraction of the circle occupied by the sector is

\frac{3}{8}

, so the arc length is

\frac{3}{8}

of the circle's circumference

2\pi r

Formulate the perimeter expression for sector

AOB

Perimeter =

2r + \frac{3}{4}\pi r

The boundary of a sector includes two straight radii of length

r

plus the curved arc length

L

Equate to given perimeter and solve for

r

r = 12

Comparing

2r + \frac{3}{4}\pi r

24 + 9\pi

, setting

2r = 24

yields

r = 12

, which also satisfies

\frac{3}{4}\pi (12) = 9\pi

Key Concept

Perimeter of a circle sector and fractional relationship between sector area, central angle, and arc length.

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