Question

what is the area of the sector that is not shaded?
12π units²
24π units²
120π units²
144π units²
(image of a circle with center l, radius 12, shaded sector with central angle 60° between points k, l, m)

Explanation:

Step1: Calculate the area of the whole circle

The formula for the area of a circle is \(A = \pi r^{2}\). Given \(r = 12\), then \(A=\pi\times12^{2}=144\pi\).

Step2: Find the central angle of the un - shaded sector

The total central angle of a circle is \(360^{\circ}\). The central angle of the shaded sector is \(60^{\circ}\), so the central angle of the un - shaded sector \(\theta=360^{\circ}- 60^{\circ}=300^{\circ}\).

Step3: Calculate the area of the un - shaded sector

The formula for the area of a sector is \(A_{sector}=\frac{\theta}{360}\times\pi r^{2}\). Substitute \(\theta = 300^{\circ}\) and \(r = 12\) into the formula: \(A_{un - shaded}=\frac{300}{360}\times144\pi\). Simplify \(\frac{300}{360}=\frac{5}{6}\), then \(A_{un - shaded}=\frac{5}{6}\times144\pi = 120\pi\).

Answer:

\(120\pi\) units\(^{2}\) (the third option)