Question

An exterior angle of a regular polygon is one fourth of its interior angle. Find the number of sides in the polygon.

Answer 1

Let $x$ be the exterior angle of a regular polygon and $y$ be the corresponding interior angle of a polygon.

Given that exterior angle is one fourth of its interior angle.

$\Rightarrow x=\frac{y}{4}$ ...$\left(1\right)$

Since $x$ and $y$ are exterior and corresponding interior angles, we have,

$x+y={180}^{\circ }$

$\Rightarrow \frac{y}{4}+y={180}^{\circ }$

$\Rightarrow \frac{5y}{4}={180}^{\circ }$

$\Rightarrow y=\frac{{180}^{\circ }\times 4}{5}={144}^{\circ }$

$\Rightarrow x={180}^{\circ }-y={180}^{\circ }-{144}^{\circ }={36}^{\circ }$

If each exterior angle of a regular polygon is $A^{\circ }$ then the number of sides in the polygon$=\frac{{360}^{\circ }}{A}$

Since exterior angle is ${36}^{\circ }$, the number of sides in the polygon$=\frac{{360}^{\circ }}{{36}^{\circ }}=10$