Chapter 5.6, Problem 51E

Solution

To find: The area of a regular polygon with $n$ sides inscribed in a circle of radius $r$ .

The area of the regular polygon in terms of $n$ and $r$ is $\frac{n{r}^2}{2}\sin \frac{360}{n}$ .

Given information:

The radius of the circle is $r$ and the length of the side of the regular polygon is $n$

Formula used:

The formula to find the area of a triangle with two sides and one included angle given is:

  $A_{triangle}=\frac{1}{2}\times \left(\text{product}\text{of}\text{sides}\right)\times \left(\text{sine}\text{of}\text{included}\text{angle}\right)$

Area of a regular polygon with $n$ sides is:

  $A_{polygon}=n\times A_{triangle}$

Calculation:

A regular polygon is made of $n$ isosceles triangle. The length of each side of the isosceles triangle shall be $n$ and the included angle will be $\frac{360}{n}$ .

Substitute $n^2$ for product of sides and $\sin \frac{360}{n}$ for sine of included angle in $A_{triangle}$ to find the area of triangle.

  $\begin{array}{c}{A}_{triangle}=\frac{1}{2}\times \left(\text{product}\text{of}\text{sides}\right)\times \left(\text{sine}\text{of}\text{included}\text{angle}\right)\\ =\frac{1}{2}\times n\times \sin \frac{360}{n}\end{array}$

Substitute $\frac{1}{2}\times n\times \sin \frac{360}{n}$ for $A_{triangle}$ in $A_{polygon}=n\times A_{triangle}$ to calculate the area of regular polygon of $n$ sides

  $\begin{array}{c}{A}_{polygon}=n\times A_{triangle}\\ =\frac{n{r}^2}{2}\sin \frac{360}{n}\end{array}$

Therefore, area of the regular polygon in terms of $n$ and $r$ is $\frac{n{r}^2}{2}\sin \frac{360}{n}$