Chapter 7, Problem 11AE

To determine

To find: The number of sides of the polygon.

Answer to Problem 11AE

15 sides

Explanation of Solution

Given information: In a regular polygon, the ratio of the measure of an exterior angle to the measure of an interior angle is $2:13$ .

Calculation:

Let the polygon has n sides.

Then, the measure of an exterior angle $=\frac{360}{n}$

The measure of an interior angle $=\frac{(n-2)180}{n}$

According to the problem,

  $\begin{array}{l}\frac{\frac{360}{n}}{\frac{(n-2)180}{n}}=\frac{2}{13}\\ \Rightarrow \frac{360}{n}\times \frac{n}{(n-2)180}=\frac{2}{13}\\ \Rightarrow \frac{360}{(n-2)180}=\frac{2}{13}\\ \Rightarrow \frac{2}{(n-2)}=\frac{2}{13}\\ \Rightarrow 2\times 13=2\times (n-2)\\ \Rightarrow n-2=13\\ \Rightarrow n=13+2\\ \Rightarrow n=15\end{array}$

Thus, the polygon has 15 sides.