Question

Find the number of sides of a regular polygon if each of its interior angles is
(i) ${90^ \circ }$
(ii) ${108^ \circ }$
(iii) ${165^ \circ }$

Answer 1

Hint:
We will use the formula of measurement of each interior angle of a polygon, which is $\theta = \dfrac{{n - 2}}{n} \times {180^ \circ }$, where $\theta $ is the interior angle and $n$ is the number of sides of the polygon. We will substitute the given values of angles to find the number of corresponding sides.

Complete step by step solution:
We have to find the number of sides when each angle of a polygon is ${90^ \circ }$
We know that measure of each interior angle of a regular polygon with $n$ sides is given as,
$\theta = \dfrac{{n - 2}}{n} \times {180^ \circ }$
Substitute ${90^ \circ }$ for $\theta $ and solve the equation to determine the value of $n$
$
  {90^ \circ } = \dfrac{{n - 2}}{n} \times {180^ \circ } \\
   \Rightarrow \dfrac{{n - 2}}{n} = \dfrac{1}{2} \\
$
Cross-multiply and rearrange the above equation,
$
  2\left( {n - 2} \right) = n \\
   \Rightarrow 2n - 4 = n \\
   \Rightarrow n = 4 \\
$
Therefore, a regular polygon with each interior angle of ${90^ \circ }$will have 4 sides.

In part (ii), we have to find the number of sides when each angle of a polygon is ${108^ \circ }$
We know that measure of each interior angle of a regular polygon with $n$ sides is given as,
$\theta = \dfrac{{n - 2}}{n} \times {180^ \circ }$
Substitute ${108^ \circ }$ for $\theta $ and solve the equation to determine the value of $n$
$
  {108^ \circ } = \dfrac{{n - 2}}{n} \times {180^ \circ } \\
   \Rightarrow \dfrac{{n - 2}}{n} = \dfrac{3}{5} \\
$
Cross-multiply and rearrange the above equation,
$
  5\left( {n - 2} \right) = 3n \\
   \Rightarrow 5n - 10 = 3n \\
   \Rightarrow 2n = 10 \\
   \Rightarrow n = 5 \\
$
Therefore, a regular polygon with each interior angle of ${108^ \circ }$will have 5 sides.

In part (iii), we have to find the number of sides when each angle of a polygon is ${165^ \circ }$
We know that measure of each interior angle of a regular polygon with $n$ sides is given as,
$\theta = \dfrac{{n - 2}}{n} \times {180^ \circ }$
Substitute ${165^ \circ }$ for $\theta $ and solve the equation to determine the value of $n$
$
  {165^ \circ } = \dfrac{{n - 2}}{n} \times {180^ \circ } \\
   \Rightarrow \dfrac{{n - 2}}{n} = \dfrac{{11}}{{12}} \\
$
Cross-multiply and rearrange the above equation,
$
  12\left( {n - 2} \right) = 11n \\
   \Rightarrow 12n - 24 = 11n \\
   \Rightarrow n = 24 \\
$

Therefore, a regular polygon with each interior angle of ${165^ \circ }$will have 24 sides.

Note:
A regular polygon is a polygon with all equal sides and angles. The polygon with the least number of sides is an equilateral triangle. There is no polygon with 2 sides. Also, the sum of all exterior angles of a polygon is ${360^ \circ }$