Chapter 5, Problem 72RE

Solution

To calculate: The area inside the pentagon and outside the circle if a circle is inscribed in a regular pentagon.

The area inside the pentagon and outside the circle is $33.5{\text{cm}}^2$ .

Given information: A circle is inscribed in a regular pentagon. The sides of regular pentagon are $12$ centimeters.

Formula used: The formula to find the area of a circle is $A=\pi r^2$ .

The formula to find the area of regular pentagon is given by $A=\frac{1}{2}\left(\text{perimeter}\right)\left(\text{apothem}\right)$ .

Calculation:

Let the radius of the circle be $r$ . Draw the figure of a circle inscribed inside a regular pentagon as shown below.

  

Figure (1)

Calculate the radius $r$ .

  $\begin{array}{c}\tan 36°=\frac{6}{r}\\ r=\frac{6}{\tan 36°}\\ r\approx 8.258\end{array}$

Substitute $8.258$ for $r$ in the formula of area of circle.

  $\begin{array}{c}{A}_{\text{circle}}=\pi \cdot {\left(8.258\right)}^2\\ =3.1415{\left(8.258\right)}^2\\ =214.24{\text{cm}}^2\end{array}$

Calculate the area of regular pentagon.

  $\begin{array}{c}{A}_{\text{pentagon}}=\frac{1}{2}\left(\text{12}\cdot \text{5}\right)\left(8.258\right)\\ =6\left(5\right)\left(8.258\right)\\ =247.74{\text{cm}}^2\end{array}$

To find the area inside the pentagon and outside the circle, subtract the area of circle from the area of the pentagon.

  $\begin{array}{c}\text{Area}=247.74{\text{cm}}^2-214.24{\text{cm}}^2\\ =33.5{\text{cm}}^2\end{array}$

Thus, the area inside the pentagon and outside the circle is $33.5{\text{cm}}^2$ .