Chapter 12.3, Problem 17E

To determine

To find :themeasure of an interior angle and an exterior angle for the regular $15-gon$

Answer to Problem 17E

The measure of an interior angle is ${156}^{\circ }$ and an exterior angle is ${24}^{\circ }$ for the regular $15-gon$ .

Explanation of Solution

Given : the polygon provided in the question,

  $15-gon$ .

Formula used: the measure of an interior angle of a regular n -gon is $\frac{\left(n-2\right)·{180}^{\circ }}{n}$ .

Each exterior angle for a regular n-gon has measure equal to 360/n degrees.

Calculation: for $15-gon$

  $n=15$ , so substitute the n by 15 in the formula, then subtract and multiply and then divide.

the measure of an interior angle of a regular $15-gon$ = $\begin{array}{l}\frac{\left(15-2\right)·{180}^{\circ }}{15}\text{ [substitute n by 15]}\\ \frac{13·{180}^{\circ }}{15}\text{ [subtract]}\\ {156}^{\circ }\text{ [multiply and divide]}\end{array}$ .

The measure of an exterior angle of a regular $15-gon$ = $\begin{array}{l}\frac{{360}^{\circ }}{15}\\ {24}^{\circ }\text{ [divide]}\end{array}$

Hence,The measure of an interior angle is ${156}^{\circ }$ and an exterior angle is ${24}^{\circ }$ for the regular $15-gon$ .