Question

a triangle is placed in a semicircle with a radius of 3 yd, as shown below. find the area of the shaded region.
use the value 3.14 for π, and do not round your answer. be sure to include the correct unit in your answer.

Explanation:

Step1: Calculate the area of the semicircle

The formula for the area of a circle is \(A = \pi r^{2}\). For a semicircle, the area \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Given \(r = 3\) yd and \(\pi=3.14\), we have \(A_{semicircle}=\frac{1}{2}\times3.14\times3^{2}=\frac{1}{2}\times3.14\times9 = 14.13\) \(yd^{2}\).

Step2: Calculate the area of the triangle

The base of the triangle is \(b = 2r=6\) yd (diameter of the semicircle) and the height \(h = r = 3\) yd. The formula for the area of a triangle is \(A_{triangle}=\frac{1}{2}bh\). So \(A_{triangle}=\frac{1}{2}\times6\times3=9\) \(yd^{2}\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{semicircle}-A_{triangle}\). Substitute the values: \(A=14.13 - 9=5.13\) \(yd^{2}\).

Answer:

\(5.13\) \(yd^{2}\)