Question

The ratio of number of sides of two regular polygons are as $5:4$ and the difference of there exterior angles is $9^o$. Find the number of sides of both the polygon.

Answer 1

Let $n$ be the Greatest Common Divisor (GCD) of the numbers under the question.

Then one polygon has $5n$ sides, while the other has $4n$ sides.

It is well known fact that the sum of exterior angles of each (convex) polygon is ${360}^o$.

So, the exterior angle of the regular $5n$-sided polygon is $\frac{{360}^o}{5n}$.

Similarly, the exterior angle of the regular $4n$-sided polygon is $\frac{{360}^o}{4n}.$

The difference between the corresponding exterior angles is $9^o.$

$\Rightarrow$ $\frac{{360}^o}{4n}-\frac{{360}^o}{5n}=9^o$

$\Rightarrow$ $\frac{1}{4n}-\frac{1}{5n}=\frac{9^o}{{360}^o}$

$\Rightarrow$ $\frac{5n-4n}{20n^2}=\frac{1}{40}$

$\Rightarrow$ $\frac{n}{20n^2}=\frac{1}{40}$

$\Rightarrow$ $\frac{1}{n}=\frac{1}{2}$

$\therefore$ $n=2$

$\Rightarrow$ Number of sides $=4n=2\times 4=8$ and $5n=2\times 5=10$.