If the difference between the exterior angle of a 'n' sided regular polygon and an (n + 1) sided regular polygon is 12°, find the value of n.

#### Solution

We know that sum of exterior angles of a polygon = 360°

Each exterior angle of a regular polygon of 360°

n sides = `360^circ/"n"`

and exterior angle of the regular polygon of

(n + 1) sides = `360^circ/("n" + 1)`

`therefore 360^circ/"n" - 360^circ/("n" + 1) = 12`

`=> 360 [1/"n" - 1/("n" + 1)] = 12`

`=> 360 [("n" + 1 - "n")/("n"("n" + 1))] = 12`

`=> (30 xx 1)/("n"^2 + "n") = 12`

`=> 12 ("n"^2 + "n") = 360^circ`

⇒ n^{2} + n = 36 (Dividing by 12)

⇒ n^{2} + n - 30 = 0

⇒ n^{2} + n - 30 = 0

⇒ n^{2} + 6n - 5n - 30 = 0 ...{∵ -30 = 6 × (-5), 1 = 6 - 5}

⇒ n(n + 6) - 5(n + 6) = 0

⇒ (n + 6)(n + 5) = 0

Either n + 6 = 0, then n = -6 which is not possible being negative

orn - 5 = 0 then n = 5

Hence n = 5.