In a circle with center O, the radius is 6. Points A and B lie on the circle such that the… — SAT

In a circle with center $O$ , the radius is $6$ . Points $A$ and $B$ lie on the circle such that the area of the sector $AOB$ is $12$ . What is the length of the minor arc $AB$ ?

Answer: 4

Answer

The length of the minor arc

AB

is 4.

The area of a sector with central angle

\theta

in radians is

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

. Setting

A = 12

and

r = 6

, we get

12 = \frac{1}{2}(6)^2\theta

, which simplifies to

12 = 18\theta

, and thus

\theta = \frac{2}{3}

radians. The length of the arc is

s = r\theta = 6 \left(\frac{2}{3}\right) = 4

Step-by-Step Solution

Find the central angle

\theta

in radians using the sector area formula.

\theta = \frac{2}{3}

The area of a sector is given by

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

, so substituting

A = 12

and

r = 6

gives

12 = 18\theta

, which yields

\theta = \frac{2}{3}

Calculate the arc length

s

using the formula

s = r\theta

s = 4

Substituting

r = 6

and

\theta = \frac{2}{3}

into the arc length formula gives

s = 6 \left(\frac{2}{3}\right) = 4

Key Concept

Calculating arc length from sector area and radius using radian measures

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