Question

If the individual measure of an angle in a regular polygon is ${156^0}$, how many sides does the polygon have?

Answer 1

Hint:To find the number of sides of a regular polygon if one of its interior angle is given, use the formula for finding the measure of each interior angle in a regular polygon if it has $n$ sides, this formula is given as $\theta = \dfrac{{{{180}^0}(n - 2)}}{n},\;{\text{where}}\;\theta $ is the measure of interior angle of the regular polygon of $n$ sides.

Formula used:
Formula to find interior angle of a regular polygon:
$\theta = \dfrac{{{{180}^0}(n - 2)}}{n}$

Complete step by step answer:
Here we are given the measure of an angle of the regular polygon which is equals to ${156^0}$, a regular polygon has equal interior angles, that means all angles are of ${156^0}$.Now, to find number of sides the polygon have, we will use the following formula of calculating interior angle of a regular polygon of $n$ sides $\theta = \dfrac{{{{180}^0}(n - 2)}}{n},\;{\text{where}}\;\theta $ is the measure of interior angle
We have $\theta = {156^0}$
$ \Rightarrow {156^0} = \dfrac{{{{180}^0}(n - 2)}}{n}$
Solving this for $n$, we will get
${156^0}n = {180^0}n - {360^0} \\
\Rightarrow {180^0}n - {156^0}n = {360^0} \\
\Rightarrow {24^0}n = {360^0} \\
\Rightarrow n = \dfrac{{{{360}^0}}}{{{{24}^0}}} \\
\therefore n = 15 \\ $
Therefore the regular polygon whose angle is ${156^0}$ has $15$ sides.

Note:We have considered that the angle given is the interior angle of the regular polygon, otherwise to find the number of sides when exterior angle of a regular polygon is given, remember that sum of all the exterior angles of a regular polygon equals ${360^0}$, and also we know that a regular polygon has equal exterior angles. So if $x$ is the given exterior angle and assume that there are $n$ number of sides it has, then we can write $nx = {360^0}$, from here we can find the number of sides.