Chapter 11.5, Problem 37SPR

To determine

To Find: The area of the shaded region.

Answer to Problem 37SPR

The area of the shaded region is $37.4{\text{in}}^2$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the area of the circle is,

  $\begin{array}{c}A=\pi r^2\\ =\pi {\left(7\right)}^2\\ =49\pi \end{array}$

Then, the formula for the area of the regular polygon with the radius $r$ is,

  $S=\frac{1}{2}n{r}^2\sin \left(\frac{360°}{n}\right)$

The area of the isosceles triangle is of the form,

  $\begin{array}{c}A=\frac{1}{2}ab\sin c\\ =\frac{1}{2}n{r}^2\sin \left(\frac{360°}{n}\right)\end{array}$

Then, the area of the n gon is,

  $A=n\left(\frac{1}{2}{r}^2\sin \left(\frac{360°}{n}\right)\right)$

Substitute the values in the formula $S=\frac{1}{2}n{r}^2\sin \left(\frac{360°}{n}\right)$ as,

  $\begin{array}{c}S=\frac{1}{2}\left(5\right){\left(7\right)}^2\sin \left(\frac{360°}{5}\right)\\ =\frac{1}{2}\left(5\right)\left(49\right)\left(\sin 72°\right)\\ =116.5\end{array}$

Thus, the area of the shaded region is,

  $\begin{array}{c}A=49\pi -116.5\\ =37.4{\text{in}}^2\end{array}$