Chapter 11.4, Problem 23PPS

To determine

To find: The area of the shaded region.

Answer to Problem 23PPS

The area of the shaded region is $11.8{\text{in}}^2$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

The area of the shaded region is equal to the difference of the area by subtracting the area of the hexagon from the area of the circle.

Since, the hexagon is inside the circle than the central angle is divided into six part so the triangle formed by the side and radius is the equilateral triangle.

The length of the side of the triangle is,

  $\begin{array}{l}\cos 30°=\frac{h}{x}\\ \frac{\sqrt{3}}{2}=\frac{h}{x}\\ x=\frac{2}{\sqrt{3}}\left(4\right)\\ x=\frac{8}{\sqrt{3}}\text{in}\end{array}$

The area of the circle having radius is,

  $\begin{array}{c}A=\pi r^2\\ =\pi {\left(\frac{8\sqrt{3}}{3}\right)}^2\\ =67{\text{in}}^2\end{array}$

Then, the area of the pentagon is,

  $\begin{array}{c}A=6\left(\frac{1}{2}bh\right)\\ =6\left(\frac{8}{\sqrt{3}}\left(4\right)\right)\\ =55.2{\text{in}}^2\end{array}$

Thus, the area of the shaded region is,

  $67{\text{in}}^2-55.2{\text{in}}^2=11.8{\text{in}}^2$