Chapter 11.4, Problem 32PPS

To determine

To find: The area of the shaded region.

Answer to Problem 32PPS

The required area of the regular triangle is $52{\text{in}}^2$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the formula for the area of the regular polygon with the side length of s is,

  $S=\frac{n}{4\tan \left(\frac{180°}{n}\right)}{s}^2$

Consider the isosceles triangle with the vertex at the centre of the pentagon with the segment that bisects the central angle and the sides of the polygon to form the right triangle is shown in Figure 2

  

Figure 2

Find the tangent ratio with the help of the apothem as,

  $\begin{array}{l}\tan \left(\frac{180°}{n}\right)=\frac{\frac{s}{2}}{a}\\ a=\frac{\frac{s}{2}}{\tan \left(\frac{180°}{n}\right)}\end{array}$

The area with the help of side length and the apothem is obtained as,

  $\begin{array}{c}A=\frac{1}{2}aP\\ =\frac{1}{2}\left(\frac{\frac{s}{2}}{\tan \left(\frac{180°}{n}\right)}\right)ns\\ =\frac{n}{4\tan \left(\frac{180°}{n}\right)}{s}^2\end{array}$

The area of the regular 3 gon is calculated as,

  $\begin{array}{c}S=\frac{3}{4\tan \left(\frac{180°}{3}\right)}{\left(6\right)}^2\\ =\frac{3\left(36\right)}{4\tan 60°}\\ \approx 25.588\end{array}$

The area of the regular 4 gon is calculated as,

  $\begin{array}{c}S=\frac{4}{4\tan \left(4\right)}{\left(6\right)}^2\\ =\frac{4\left(36\right)}{4\tan 45°}\\ \approx 36\end{array}$

The area of the regular 5 gon is calculated as,

  $\begin{array}{c}S=\frac{5}{4\tan \left(5\right)}{\left(5\right)}^2\\ =\frac{5\left(36\right)}{4\tan 36°}\\ \approx 61.937\end{array}$

The area of the regular 6 gon is calculated as,

  $\begin{array}{c}S=\frac{6}{4\tan \left(6\right)}{\left(5\right)}^2\\ =\frac{6\left(36\right)}{4\tan 36°}\\ \approx 93.53\end{array}$

Then, the area of the composite figure is,

  $\begin{array}{c}A=6\text{-gon}-5\text{-gon}+4\text{-gon}-3\text{-gon}\\ =\text{95}\text{.53}-\text{61}\text{.937}+\text{36}-\text{25}\text{.588}\\ =52{\text{in}}^2\end{array}$