In a circle with center O, the length of minor arc AB is 3\pi and the area of sector AOB i… — SAT

In a circle with center $O$ , the length of minor arc $AB$ is $3\pi$ and the area of sector $AOB$ is $18\pi$ . What is the circumference of the circle?

A
$12\pi$
B
$18\pi$
$24\pi$ Answer
D
$36\pi$

Answer

The circumference of the circle is

24\pi

The correct answer is the option containing

24\pi

. By using the relationship

A = \frac{1}{2}rs

, where

A

is the sector area,

r

is the radius, and

s

is the arc length, we substitute the given values to get

18\pi = \frac{1}{2}r(3\pi)

. Solving for the radius yields

r = 12

. Substituting this radius into the circumference formula

C = 2\pi r

gives

2\pi(12) = 24\pi

Step-by-Step Solution

Write the formulas for arc length

s

and sector area

A

in terms of radius

r

and central angle

\theta

in radians.

s = r\theta = 3\pi

and

A = \frac{1}{2}r^2\theta = 18\pi

This sets up the system of equations using the given geometric properties.

Express the sector area formula in terms of arc length by substituting

s = r\theta

into

A = \frac{1}{2}r(r\theta)

A = \frac{1}{2}rs

, which becomes

18\pi = \frac{1}{2}r(3\pi)

This simplifies the relationship to a single equation with one variable,

r

Solve the equation

18\pi = 1.5\pi r

for the radius

r

r = 12

Finding the radius is necessary to calculate the circumference of the circle.

Substitute

r = 12

into the circumference formula

C = 2\pi r

C = 2\pi(12) = 24\pi

This provides the final circumference value requested by the question.

Key Concept

Relationship between arc length, sector area, and circumference in circle geometry

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