Question

If the sum of the interior angles of a regular polygon is 1620°, what is the measure of one exterior angle. Round to the nearest tenth.

Answer 1

`\textit{sum of all interior angles in a polygon}\\\\ S =180(n-2) \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ S =1620 \end{cases}\implies 1620=180(n-2) \\\\\\ \cfrac{1620}{180}=n-2\implies 9=n-2\implies \underline{11=n} \\\\[-0.35em] ~\dotfill`

`\textit{sum of all exterior angles in a regular polygon}\\\\ n\theta = 360 \begin{cases} n=\stackrel{number~of}{sides}\\ \theta =\stackrel{angle~in}{degrees}\\[-0.5em] \hrulefill\\ \underline{n=11} \end{cases}\implies 11\theta =360\implies \theta =\cfrac{360}{11}\implies \theta \approx 32.7^o`

Answer 2

Interior angles of a regular polygon

A regular polygon is a figure in which all the sides have the same length that is, they are equal and all the interior angles are of the same measure.

The formula to calculate the interior angles of any regular polygon is:

( n - 2 ) 180°

This regular polygon is a hendecagon.

A hendecagon is a polygon that has 9 equal sides.

If the sum of the interior angles is 1620°, then

( n - 2 ) 180° = 1620

As a second step, we divide by 180:

( n-2 ) = 1620 ➗ 180

We divide

n - 2 = 9

We change the direction, as the sign is negative, in the change of direction it starts to add.

n = 9 + 2

We add both numbers.

n = 11

Answer: the measure of the exterior angle of the regular polygon (hendecagon) is 11. ✅