Question

The ratio between the number of sides of two regular polygons is 3 : 4 and the ratio between the sum of their interior angles is 2 : 3. Find the number of sides in each polygon.

Answer 1

Ratio of sides of two regular polygons = 3 : 4

Let sides of first polygon = 3n

and sides of second polygon = 4n

$\therefore$ Sum of interior angles of first polygon $=(2\times 3n-4)\times {90}^o=(6n-4)\times {90}^o$

and sum of interior angle of second polygon

$=(2\times 4n-4)\times {90}^o=(8n-4)\times {90}^o\therefore \frac{(6n-4)\times {90}^o}{(8n-4)\times {90}^o}=\frac{2}{3}\Rightarrow \frac{6n-4}{8n-4}=\frac{2}{3}\Rightarrow 18n-12=16n-8\Rightarrow 18n-16n=-8+12\Rightarrow 2n=4\Rightarrow n=2$

$\therefore$ No. of sides of first polygon $=3n=3\times 2=6$

and no. of sides of second polygon

$=4n=4\times 2=8$