Chapter 12.3, Problem 34E

To determine

To state:thepolygon for which the sum of the measures of the interior angles is ${2160}^{\circ }$ .

Answer to Problem 34E

the polygon is $14-gon$ for which the sum of the measures of the interior angles is ${2160}^{\circ }$ .

Explanation of Solution

Given : the sum of the measure of the interior anglesis ${2160}^{\circ }$ .

Formula used:

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

Calculation:

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

  $\begin{array}{l}\left(n-2\right)·{180}^{\circ }={2160}^{\circ }\\ \frac{\left(n-2\right)·{180}^{\circ }}{{180}^{\circ }}=\frac{{2160}^{\circ }}{{180}^{\circ }}\text{ [divide each side by }{180}^{\circ }]\\ \left(n-2\right)=12\\ n-2+2=12+2\text{ [add both side by 2]}\\ n=14\end{array}$

So the polygon is $14-gon$ .

Hence,the polygon is $14-gon$ for which the sum of the measures of the interior angles is ${2160}^{\circ }$