Question

Determine the number of sides of a polygon whose exterior and interior angles are in the ratio 1:5.

Answer 1

Ratio in exterior angle and interior angles of a regular polygon = 1:5
But sum of interior and exterior angles = ${180}^{\circ }$ (Linear pair)
$\therefore$ Exterior angle = $\frac{{180}^{\circ }\times 1}{1+5}=\frac{{180}^{\circ }\times 1}{6}={30}^{\circ }$
and interior angles = $\frac{{180}^{\circ }\times 5}{6}={150}^{\circ }$
Let number of sides of the polygon = n
$\therefore \frac{2n-4}{n}\times {90}^{\circ }={150}^{\circ }\Rightarrow \frac{2n-4}{n}=\frac{150}{90}=\frac{5}{3}$
By cross multiplication:
$6n-12=5n\Rightarrow 6n-5n=12\Rightarrow n=12$
$\therefore$ Number of sides of polygon is 12.