A board game spinner has a pointer of length 6\text{ inches} that pivots at the center of … — ACT

A board game spinner has a pointer of length $6\text{ inches}$ that pivots at the center of the circular board. If the sector of the board representing 'Lose a Turn' has a central angle of $\frac{\pi}{3}$ radians, what is the area, in square inches, of this sector?

A
$2\pi$
B
$\frac{3\pi}{2}$
$6\pi$ Answer
D
$12\pi$
E
$36\pi$

Answer

6\pi

square inches

The area of a sector with a radius

r

and a central angle

\theta

(measured in radians) is determined by the formula

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

. Given that the radius

r = 6

and the central angle

\theta = \frac{\pi}{3}

, substituting these values into the formula gives

A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 6\pi

square inches.

Step-by-Step Solution

Identify the radius and the central angle of the sector from the problem description.

Radius

r = 6\text{ inches}

, and central angle

\theta = \frac{\pi}{3}\text{ radians}

These values are the direct inputs required for the circle sector formulas.

Use the sector area formula in radians,

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

, to calculate the area.

A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 18\left(\frac{\pi}{3}\right) = 6\pi

This formula scales the total area of the circle by the fraction represented by the radian angle relative to

2\pi

radians.

Key Concept

Calculating the area of a circle sector using radian measure

Estimated Time:1m 0s

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