Chapter 11.5, Problem 73E

To determine

To Find: The area of a regular pentagon that is inscribed in a circle with a diameter of 7.3 centimetres.

Answer to Problem 73E

The area of the pentagon is $31.7{\text{cm}}^2$ .

Explanation of Solution

Consider the radius of the circle is,

  $\frac{7.3}{2}=3.65$

The angle measure of $\alpha$ is,

  $\begin{array}{c}\alpha =\frac{360°}{10}\\ =36°\end{array}$

Then, the value of the cos $36°$ is,

  $\begin{array}{l}\cos 36°=\frac{a}{3.65}\\ a=3.65\left(\cos 36°\right)\\ a=2.95\text{cm}\end{array}$

Then,

  $\begin{array}{c}CE=\sqrt{{\left(3.65\right)}^2-{\left(2.95\right)}^2}\\ =2.15\text{cm}\end{array}$

The required diagram is shown in Figure 1

  

Figure 1

From the above figure,

  $\begin{array}{c}CD=2CE\\ =4.3\text{cm}\end{array}$

Then, the area of the triangle is,

  $\begin{array}{c}\frac{1}{2}a\left(CD\right)=\frac{1}{2}\left(4.3\right)\left(2.95\right)\\ =6.34{\text{cm}}^2\end{array}$

Thus, the area of the pentagon is,

  $5\left(6.34\right)=31.7{\text{cm}}^2$