Question

Solve the following :
The ratio between exterior angle and interior angle of a regular polygon is $1:5$. Find the number of sides of the polygon.

Answer 1

From the question it is given that, The ratio between an exterior angle and the interior angle of a regular polygon is 1: 5
Let us assume exterior angle be $y$ And interior angle be $5y$
We know that, sum of interior and exterior angle is equal to ${180}^{\circ }$,
$\begin{array}{c}y+5y={180}^{\circ }\\ 6y={180}^{\circ }\\ y={180}^{\circ }/6\\ y={30}^{\circ }\end{array}$
the number of sides in the polygon The number of sides of a regular polygon whose each interior angles has a measure of
${150}^{\circ }$
Let us assume the number of sides of the regular polygon be $n$,
Then, we know that ${150}^{\circ }=\left(\right(2n-4\left)/n\right)\times {90}^{\circ }$
$\begin{array}{c}{150}^{\circ }/{90}^{\circ }=(2n-4)/n\\ 5/3=(2n-4)/n\end{array}$
By cross multiplication,$3(2n-4)=5n$
$6n-12=5n$
By transposing we get,
$\begin{array}{c}6n-5n=12\\ n=12\end{array}$
Therefore, the number of sides of a regular polygon is 12 .