In circle O, a sector with a central angle of 60^\circ has an area of 6\pi. In circle P, a… — GRE General Test

In circle $O$ , a sector with a central angle of $60^\circ$ has an area of $6\pi$ . In circle $P$ , a sector with a central angle of $120^\circ$ has an arc length equal to the arc length of the sector in circle $O$ . What is the area of the sector in circle $P$ ?

A
$12\pi$
B
$6\pi$
$3\pi$ Answer
D
$9\pi$
E
$\frac{3\pi}{2}$

Answer

3\pi

The sector area of

3\pi

is correctly calculated by first determining the radius of circle

O

(

r_1 = 6

), using it to find the arc length (

2\pi

), setting that as the arc length for the sector in circle

P

to find its radius (

r_2 = 3

), and finally computing its sector area

\frac{120^\circ}{360^\circ} \cdot \pi (3^2) = 3\pi

Step-by-Step Solution

Find the radius of circle

O

Radius

r_1 = 6

The sector area formula is

A = \frac{\theta}{360^\circ} \cdot \pi r^2

. Given

A = 6\pi

and

\theta = 60^\circ

, we have

\frac{60^\circ}{360^\circ} \cdot \pi r_1^2 = 6\pi \Rightarrow \frac{1}{6} r_1^2 = 6 \Rightarrow r_1^2 = 36 \Rightarrow r_1 = 6

Calculate the arc length of the sector in circle

O

Arc length

L = 2\pi

The arc length formula is

L = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

\theta = 60^\circ

and

r_1 = 6

, we get

L = \frac{1}{6} \cdot 2\pi (6) = 2\pi

Find the radius of circle

P

Radius

r_2 = 3

The sector in circle

P

has an arc length equal to

2\pi

and a central angle of

120^\circ

. Using

L = \frac{120^\circ}{360^\circ} \cdot 2\pi r_2

, we get

2\pi = \frac{1}{3} \cdot 2\pi r_2 \Rightarrow r_2 = 3

Calculate the area of the sector in circle

P

Sector area

= 3\pi

Using the sector area formula for circle

P

A_2 = \frac{120^\circ}{360^\circ} \cdot \pi r_2^2 = \frac{1}{3} \cdot \pi (3^2) = 3\pi

Key Concept

Relationship between central angles, radii, arc lengths, and sector areas in circles

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