Find the number of sides of a regular polygon if the exterior angle is one-third of its interior angle.

Given :

The exterior angle of a regular polygon is one-third of its interior angle.
 

To do :

We have to find the number sides of the polygon.

Solution :

Let the number of sides of the regular polygon be 'n'.

The exterior angle of a regular polygon with n sides $= \frac{360}{n}$

The interior angle of a regular polygon with n sides $=180 -  \frac{360}{n}$

Here, the exterior angle is one-third of its interior angle.

$\frac{360}{n} = \frac{1}{3}(180 -  \frac{360}{n})$

$\frac{360\times 3}{n} = 180 -  \frac{360}{n}$

$\frac{360\times 3}{n} = \frac{180 n - 360}{n}$

$360\times 3 =180 n - 360 $                        [n on both sides get cancelled]

$360\times 3 + 360=180 n $

Take 360 as common in LHS,

$360(3 + 1) = 180 n $

$360 \times 4 = 180 n$

Rewrite, 

$180 n = 360 \times 4$

$n = \frac{360 \times 4}{180}$

$n = 2 \times 4$                                                      $[\frac{360}{180} = 2]$

$n = 8$

Therefore, the number of sides of the regular polygon is 8.