Chapter 6.2, Problem 45E

To determine

To find: the area of regular pentagon for the given situation.

Answer to Problem 45E

  $31.7{\text{ cm}}^{\text{2}}$

Explanation of Solution

Given information:

The regular pentagon is inscribed in a circle.

Diameter of circle = 7.3cm

Calculation:

The diagram for the given condition will be as follows:

  

Let the base, height and hypotenuse of the triangle be x , y and r respectively. The hypotenuse and radius of circle are equal.

The radius of circle will be evaluated as follows:

  $\begin{array}{c}\text{Radius}\left(r\right)=\frac{\text{Diameter}}{2}\\ =\frac{7.3}{2}\text{ cm}\\ =3.65\text{ cm}\end{array}$

Thus the radius of the circle is 3.65 cm.

Look at the figure carefully; if you create two triangles as shown in the figure for each side, then the total number of triangles in the entire circle will be 10. Thus the value of $\theta$ for each triangle will be as follows:

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

The base $\left(x\right)$ of the triangle will be evaluated as follows:

  $\begin{array}{c}\frac{\text{Base}}{\text{Hypotenus}}=\sin \theta \\ \frac{x}{r}=\sin \left(36°\right)\\ \frac{x}{3.65}=\sin \left(36°\right)\left[\text{Put the value of }r\right]\\ \frac{x}{3.65}=0.587\\ x=\left(0.587\right)\left(3.65\right)\\ =2.15\text{ cm}\end{array}$

The height $\left(y\right)$ of the triangle will be evaluated as follows:

  $\begin{array}{c}\frac{\text{Perpendicular}}{\text{Hypotenus}}=\cos \theta \\ \frac{y}{r}=\cos \left(36°\right)\\ \frac{y}{3.65}=\cos \left(36°\right)\left[\text{Put the value of }r\right]\\ \frac{y}{3.65}=0.809\\ y=\left(0.809\right)\left(3.65\right)\\ =2.95\text{ cm}\end{array}$

Thus the base and height of the triangle is 2.15 cm and 2.95 cm respectively.

The area of triangle is evaluated as follows:

  $\begin{array}{c}\text{Area of }\Delta =\frac{1}{2}\left(x\right)\left(y\right)\\ =\frac{1}{2}\left(2.15\text{ cm}\right)\left(2.95\text{ cm}\right)\left[\text{Put the value of }x\text{ and }y\right]\\ =3.17{\text{ cm}}^2\end{array}$

Thus the area of triangle is $3.17{\text{ cm}}^2$ .