Question

The number of sides of two regular polygons are in the ratio $5:4$ and the difference between their interior angles is $6^{\circ }$. Find the number of sides in the two polygons.

• A. $11$, $12$
• B. $15$, $12$
• C. $12$, $14$
• D. $13$, $15$

Answer 1

Answer: (B)

We know that each interior angle of a regular polygon of $m$ sides is
$180^{\circ }$ - each exterior angle $=180^{\circ }-\frac{360^{\circ }}{m}$
$(\because$ Sum of all exterior angles is always $360^{\circ })$
Let the number of sides be $5n$ and $4n$, then each interior angle of first polygon ${\left(180-\frac{360}{5n}\right)}^{\circ }=180{\left(1-\frac{2}{5n}\right)}^{\circ }$
and each interior angle of second polygon
$={\left(180-\frac{360}{4n}\right)}^{\circ }=180{\left(1-\frac{2}{4n}\right)}^{\circ }$
It is given that $180\left(1-\frac{2}{5n}\right)-180\left(1-\frac{1}{2n}\right)=6$
$\Rightarrow n=3$.
Hence, the number of sides in the two polygons are
$5\times 3=15$ and $4\times 3=12$.