Question

The ratio between the number of sides of two regular polygons is $1:2$. The ratio of the measures of their interior angles is $3:4$. How many sides are there in each polygon?

• A. 7 and 10
• B. 5 and 15
• C. 5 and 10
• D. 10 and 12

Answer 1

The correct option is C 5 and 10

Given: The sides of the two regular polygons are in the ratio $1:2$.

Let the number of sides of the polygons be $n$ and $2n$ respectively

The interior angles of these polygons are in the ratio $3:4$.

The measure of each interior angle for a regular polygon with n sides is given by $\frac{(n-2)\times {180}^{\circ }}{n}$

Therefore, the measure of each interior angle of the given polygons will be $\frac{(n-2)\times {180}^{\circ }}{n}$ and $\frac{(2n-2)\times {180}^{\circ }}{2n}$ respectively.

Also, the ratio of the interior angles is $3:4$.

$\therefore \frac{(n-2)\times {180}^{\circ }}{n}÷\frac{(2n-2)\times {180}^{\circ }}{2n}=\frac{3}{4}$

$\Rightarrow \frac{n-2}{n-1}=\frac{3}{4}$

$\Rightarrow 4n-8=3n-3$

$\Rightarrow 4n-3n=8-3$

$\therefore n=5$

Therefore, the number of sides in each polygon is $n$ and $2n$, which is $5$ and $10$ respectively.