Question

Find the number of sides of a regular polygon if each exterior angle is equal to one third of its adjacent interior angle.

Answer 1

The sum of exterior angle, $e$ and its adjacent interior angle, $i$ is ${180}^{\circ }$:

$e+i={180}^{\circ }$

Since, each exterior angle is equal to one third its adjacent interior angle, therefore, substitute $\frac{i}{3}=e$ or $i=3e$ in the above equation.

$⟹e+3e={180}^{\circ }4e={180}^{\circ }e=\frac{{180}^{\circ }}{4}={45}^{\circ }$

We know that the measure of exterior angle is $e={\left(\frac{360}{n}\right)}^{\circ }$ where $n$ is the number of sides.

Here, it is given that the exterior angle is $e={45}^{\circ }$, therefore,

$n=\frac{{360}^{\circ }}{e}=\frac{{360}^{\circ }}{{45}^{\circ }}=8$

Hence, the number of sides is $8$.