Question

what is the area of the shaded portion of the circle? (16π−32) in² (16π−8) in² (64π−32) in² (64π−8) in²

Explanation:

Step1: Find area of sector

The sector is a quarter - circle (since the central angle is \(90^{\circ}\), and \(\frac{90^{\circ}}{360^{\circ}}=\frac{1}{4}\)) with radius \(r = 8\) in. The formula for the area of a circle is \(A=\pi r^{2}\), so the area of the sector is \(\frac{1}{4}\times\pi\times r^{2}\). Substituting \(r = 8\), we get \(\frac{1}{4}\times\pi\times8^{2}=\frac{1}{4}\times\pi\times64 = 16\pi\) \(in^{2}\).

Step2: Find area of triangle

The triangle is a right - triangle with legs equal to the radius of the circle (\(r = 8\) in each). The formula for the area of a right - triangle is \(A=\frac{1}{2}\times base\times height\). Here, base \(= 8\) in and height \(= 8\) in, so the area is \(\frac{1}{2}\times8\times8=32\) \(in^{2}\).

Step3: Find area of shaded region

The area of the shaded region is the area of the sector minus the area of the triangle. So, \(A = 16\pi-32\) \(in^{2}\).

Answer:

\((16\pi - 32)\text{ in}^2\)