Question

1. circle o with a radius of 9 is drawn below. the measure of central angle aoc is 120°. what is the area of the shaded sector of circle o? (1)6π (2)12π (3)27π (4)54π

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector of a circle is given by \( A=\frac{\theta}{360^\circ}\times\pi r^{2} \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, the unshaded sector has a central angle of \( 120^\circ \), so the shaded sector has a central angle of \( 360^\circ - 120^\circ=240^\circ \), and the radius \( r = 9 \).

Step2: Substitute the values into the formula

First, calculate \( \frac{\theta}{360^\circ}=\frac{240^\circ}{360^\circ}=\frac{2}{3} \). Then, calculate \( \pi r^{2}=\pi\times9^{2}=81\pi \). Now, multiply these two results: \( A = \frac{2}{3}\times81\pi \).

Step3: Simplify the expression

\( \frac{2}{3}\times81\pi=2\times27\pi = 54\pi \)? Wait, no, wait. Wait, maybe I made a mistake. Wait, the central angle for the shaded sector: wait, the unshaded is \( 120^\circ \), so shaded is \( 360 - 120 = 240^\circ \)? Wait, no, wait the diagram: maybe the shaded is the larger part. Wait, but let's check again. Wait, the formula for the sector area: if the central angle is \( \theta \), then area is \( \frac{\theta}{360}\pi r^2 \). Wait, maybe I misread the central angle. Wait, the problem says "the measure of central angle AOC is \( 120^\circ \)". So angle AOC is \( 120^\circ \), so the unshaded sector is \( 120^\circ \), so shaded is \( 360 - 120 = 240^\circ \). Wait, but let's compute:

\( r = 9 \), so \( \pi r^2 = 81\pi \). Then, the fraction of the circle that is shaded is \( \frac{240}{360}=\frac{2}{3} \). So \( \frac{2}{3}\times81\pi = 54\pi \)? But wait, the options are (1) \( 6\pi \), (2) \( 12\pi \), (3) \( 27\pi \), (4) \( 54\pi \). Wait, but maybe I made a mistake. Wait, no, wait: maybe the central angle for the shaded sector is \( 240^\circ \), but let's check again. Wait, the formula for sector area: \( A=\frac{\theta}{360}\pi r^2 \). So \( \theta = 240^\circ \), \( r = 9 \). So \( A=\frac{240}{360}\times\pi\times9^2=\frac{2}{3}\times81\pi = 54\pi \). Wait, but that's option (4). But wait, maybe I misread the central angle. Wait, maybe the shaded sector is the one with central angle \( 240^\circ \), so the calculation is correct.

Wait, no, wait a second. Wait, maybe the central angle for the shaded sector is \( 360 - 120 = 240 \), but let's check the options. Option (4) is \( 54\pi \), which is what we got. So let's verify:

\( \frac{240}{360}=\frac{2}{3} \), \( 9^2 = 81 \), \( \frac{2}{3}\times81 = 54 \), so \( 54\pi \). So that's correct.

Answer:

\( 54\pi \) (Option 4)