Question

The ratio between the number of sides of two regular polygons is $1:2$ and the ratio between their interior angles is $2:3$. The number of sides of these polygons is respectively.

• A. $6,12$
• B. $5,10$
• C. $4,8$
• D. $7,14$

Answer 1

Answer: (C)

$4,8$

Each interior angle $=\frac{(2n-4)\times {90}^{\circ }}{n}$
According to the question,
$\frac{\frac{(2n-4)\times {90}^{\circ }}{n}}{\frac{(2\times 2n-4)\times {90}^{\circ }}{2n}}=\frac{2}{3}$ (Ratio of number of sides is $1:2$)
$\Rightarrow \frac{(2n-4)\times 2}{4n-4}=\frac{2}{3}\Rightarrow \frac{2n-4}{4n-4}=\frac{1}{3}$
$\Rightarrow 6n-12=4n-4\Rightarrow 6n-4n=12-4=8$
$\Rightarrow 2n=8\Rightarrow n=4$
Hence, number of sides in polygons are $4$ and $8$ respectively.