Question

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

Answer 1

Ratio of the sides is 3 : 4
∴ Number of sides in each polygon is 3x and 4x.

Each interior angle of a regular polygon = `(("n" - 2) xx 180°)/"n"`

∴ Interior angle of a regular polygon of 3x sides = `((3x - 2) xx 180°)/(3x)`

And Interior angle of a regular polygon of 4x sides = `((4x - 2) xx 180°)/(4x)`

Ratio of the interior angles is 2 : 3

⇒ `{((3x - 2) xx 180°)/(3x)} : {((4x - 2) xx 180°)/(4x)}` = 2 : 3

⇒ `{((3x - 2) xx 180°)/(3x)} : {(4x)/((4x - 2) xx 180°)} = (2)/(3)`

⇒ `((3x - 2))/((4x - 2)) xx (4)/(3) = (2)/(3)`

⇒ 2(3x - 2) = (5x - 2)
⇒ 2x = 2
∴ x = 1
So, the number of sides of each of the polygons are 3 and 4.