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 $4 \times 3 = 12$.