Question

what is the area of the shaded sector of the circle?
9
120°
options: 9π units², 27π units², 81π units², 162π units²

Explanation:

Step1: Recall sector area formula

The formula for the area of a sector of a circle is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step2: Identify values

Here, the radius \( r = 9 \) and the central angle \( \theta=120^{\circ} \).

Step3: Substitute into formula

First, calculate \( r^{2}=9^{2} = 81 \). Then, substitute into the sector area formula: \( A=\frac{120^{\circ}}{360^{\circ}}\times\pi\times81 \). Simplify \( \frac{120}{360}=\frac{1}{3} \). So, \( A=\frac{1}{3}\times81\pi= 27\pi \).

Answer:

\( 27\pi \) units\(^2\) (corresponding to the option "27π units²")