Chapter 12, Problem 21CR

To determine

To find : The measure of an interior angle and the measure of an exterior angle for the regular $13-gon$ .

Answer to Problem 21CR

Therefore, the measure of an interior angle of regular $13-gon$ is ${152.30}^{\circ }$ and the measure of an exterior angle of a $13-gon$ is ${27.7}^{\circ }$ .

Explanation of Solution

Given information : Here, the polygon is regular $13-gon$ .

Formula used : All interior angles of a regular polygon are equal and the measure of an interior angle of regular n-gon is $\frac{\left(n-2\right)\times {180}^{\circ }}{n}$

All exterior angles of a regular polygon are equal and the measure of an exterior angle of regular n-gon is ${180}^{\circ }-\frac{\left(n-2\right)\times {180}^{\circ }}{n}$

Calculation :

Here, polygon is regular $13-gon$ i.e. $n=13$

Therefore, the measure of an interior angle of regular $13-gon$ is

  $\begin{array}{l}\frac{\left(13-2\right)\times {180}^{\circ }}{13}\\ \Rightarrow \frac{11\times {180}^{\circ }}{13}\\ \Rightarrow \frac{{1980}^{\circ }}{13}\\ \Rightarrow {152.30}^{\circ }\end{array}$

The measure of an exterior angle of a regular $13-gon$ is

  $\begin{array}{l}{180}^{\circ }-\frac{\left(13-2\right)\times {180}^{\circ }}{13}\\ \Rightarrow {180}^{\circ }-\frac{11\times {180}^{\circ }}{13}\\ \Rightarrow {180}^{\circ }-\frac{{1980}^{\circ }}{13}\\ \Rightarrow {180}^{\circ }-{152.30}^{\circ }\\ \Rightarrow {27.7}^{\circ }\end{array}$