Question

If the difference between an exterior angle of a regular polygon of 'n' sides and an exterior angle of another regular polygon of '(n + 1)' sides is equal to 4°; 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)`

Difference between the two exterior angles = 4°

`(360°)/"n" - (360°)/("n" + 1)` = 4°

`(90)/"n" - (90)/("n" + 1)` = 1

`(90"n" + 90 - 90"n")/("n"("n" + 1)` = 1

⇒ 90 = n² + n
⇒ n² + n - 90 = 0
⇒ n² + 10n - 9n - 90 = 0
⇒ n(n + 10) - 9(n + 10) = 0
⇒ (n + 10)(n - 9) = 0
⇒ n + 10 = 0 or n - 9 = 0
⇒ n = -10 or n = 9
Since the number of sides cannot be negative, we have n = 9.