Chapter 11.4, Problem 22PPS

To determine

To find: The area of the shaded region.

Answer to Problem 22PPS

The area of the shaded region is $238.7{\text{cm}}^2$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the area of the circle is,

  $\begin{array}{c}A=\pi r^2\\ =\pi {10}^2\\ =100\pi \\ =314.2{\text{m}}^2\end{array}$

Consider the area of the pentagon is five times the area of the triangle that is formed by the two radii and one side of the pentagon.

The vertical angle of the triangle is,

  $\frac{360°}{5}=72°$

Draw the perpendicular line from the vertex to the base of the triangle to get the height of the triangle is,

  $\begin{array}{l}\sin 36°=\frac{\frac{b}{2}}{10}\\ \sin 36°=\frac{b}{20}\\ b=11.8\end{array}$

Then, the height of the triangle is,

  $\begin{array}{l}\cos 36°=\frac{h}{10}\\ h=10\cos 36°\\ h=8.1\end{array}$

Thus, the area of the pentagon is,

  $\begin{array}{c}A=5\left(\frac{1}{2}bh\right)\\ =5\left(\frac{1}{2}\left(11.8\right)\left(8.1\right)\right)\\ =238.7{\text{cm}}^2\end{array}$