Question

The number of slides in two regular polygons are in the ratio of $5:4$ and the difference between each interior angle of the polygons is $6^{\circ }$. Then the sum number of sides are :

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 $6^o.$

$\Rightarrow$ $\frac{{360}^o}{4n}-\frac{{360}^o}{5n}=6^o$

$\Rightarrow$ ${360}^o(\frac{1}{4n}-\frac{1}{5n})=6^o$

$\Rightarrow$ $\frac{5n-4n}{20n^2}=\frac{6^o}{{360}^o}$

$\Rightarrow$ $\frac{n}{20n^2}=\frac{1}{60}$

$\Rightarrow$ $\frac{1}{n}=\frac{20}{60}$

$\Rightarrow$ $\frac{1}{n}=\frac{1}{3}$

$\therefore$ $n=3$

$\Rightarrow$ $5n=5\times 3=15$

$\Rightarrow$ $4n=4\times 3=12$

$\therefore$ The sum of number of sides $=15+12=27$