Chapter 12.3, Problem 33E

To determine

To cstate:thepolygon for which the sum of the measures of the interior angles is ${1620}^{\circ }$ .

Answer to Problem 33E

the polygon is $11-gon$ for which the sum of the measures of the interior angles is ${1620}^{\circ }$ .

Explanation of Solution

Given : the sum of the measure of the interior anglesis ${1620}^{\circ }$ .

Formula used:

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

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

  $\begin{array}{l}\left(n-2\right)·{180}^{\circ }={1620}^{\circ }\\ \frac{\left(n-2\right)·{180}^{\circ }}{{180}^{\circ }}=\frac{{1620}^{\circ }}{{180}^{\circ }}\text{ [divide each side by }{180}^{\circ }]\\ \left(n-2\right)=9\\ n-2+2=9+2\text{ [add 2 both side]}\\ n=11\end{array}$

So the polygon is $11-gon$ .

Hence, the polygon is $11-gon$ for which the sum of the measures of the interior angles is ${1620}^{\circ }$