Chapter 4.8, Problem 43E

To determine

Find the side of pentagon inscribed in a circle of radius 25 inches.

Answer to Problem 43E

The side of the pentagon is 29.4 inches

Explanation of Solution

Given:

Radius of circle = 25inches.

Now take the pentagon and break it into five equal pie slices at the centre of the circle. Each slice is a isosceles triangle. The angle at the centre being ${72}^{\circ }\left(\text{that is}\frac{{360}^{\circ }}{5}\right)$

The $\Delta AOB$ can now be split into half. Bisect the angle $\angle AOB$ and the chord $AB$ and you will

have a $\Delta ONB$ right angled at $N,$ with a base angle of ${36}^{°},$ an opposite side $\frac{c}{2},$ and a hypotenuse $r$ (radius of circle).

In right $\text{Δ}ONB$ , we will use the fact

  $\begin{array}{cc}\frac{C}{2r}& =\sin {36}^{\circ }\\ C& =2r\sin {36}^{\circ }\\ & =2\times 25\times 0.588\end{array}$

  $=29.4$ inches.

Therefore, the side of the pentagon is 29.4 inches