Chapter 5.6, Problem 26E

To determine

To find: The area of the pentagon.

Answer to Problem 26E

  $\text{Area}=192.6u^2$

Explanation of Solution

Given information: Geometry: A regular pentagon is inscribed in a circle whose radius measures 9 inches.

Calculation:

  $k=\frac{1}{2}bc\sin A$

Now we use our big trig ratios to find the apothem and the ½ side.

  $\begin{array}{l}\sin {36}^{\circ }=\frac{1/2\text{side}}{9}\\ 1/2\text{ side}=9\sin {36}^{\circ }=5.29\end{array}$

So the length of one side is twice this is 10.58 so the perimeter (since its pentagon) is:

  $5\times 10.58=52.9$

Now the find the apothem:

  $\begin{array}{l}\cos {36}^{\circ }=\frac{\text{apothem}}{9}\\ \text{apothem}=9\times \cos {36}^{\circ }=7.28\end{array}$

Now we can use the formula $A=\frac{1}{2}aP$

  $\begin{array}{l}A=0.5\times 7.28\times 52.9\\ A=192.6\end{array}$

Thus,

  $\text{Area}=192.6u^2$