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 area of semi - circle

The formula for the area of a circle is $A = \pi r^{2}$. For a semi - circle, the area $A_{semicircle}=\frac{1}{2}\pi r^{2}$. Given $r = 3$ yd and $\pi=3.14$, we have $A_{semicircle}=\frac{1}{2}\times3.14\times3^{2}=\frac{1}{2}\times3.14\times9 = 14.13$ yd².

Step2: Calculate area of triangle

The base of the triangle is the diameter of the semi - circle, so $b = 2r=6$ yd and the height $h = r = 3$ yd. The formula for the area of a triangle is $A=\frac{1}{2}bh$. Then $A_{triangle}=\frac{1}{2}\times6\times3=9$ yd².

Step3: Calculate area of shaded region

The area of the shaded region $A_{shaded}=A_{semicircle}-A_{triangle}$. Substitute the values of $A_{semicircle}$ and $A_{triangle}$: $A_{shaded}=14.13 - 9=5.13$ yd².

Answer:

$5.13$ yd²