Question

The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6°. Find the value of n.

Answer 1

Each exterior angle of a regular polygon of n sides = `(360°)/"n"`

∴ Each exterior angle of a regular polygon of (n - 1) sides = `(360°)/("n" - 1)`

∴ Each exterior angle of a regular polygon of (n + 2) sides = `(360°)/("n" + 2)`

Difference between the two exterior angles = 6°

∴ `(360°)/("n" - 1) - (360°)/("n" + 2)` = 6

⇒ `360°[("n" + 2 - "n" + 1)/(("n" - 1)("n" + 2))]` = 6°

⇒ 60 x 3 = (n - 1)(n + 2)
⇒ 180 = n² + n - 2
⇒ n² + n - 182 = 0
⇒ n² + 14n - 13n - 182 = 0
⇒ (n + 14)(n - 13) = 0
∴ n = -14 (rejected as number of sides can't be negative) or n = 13
∴ The value of n is 13.