Question

The angles of a polygon are in A.P. with common difference 5°. If the smallest angle is 120°, find the number of sides of the polygon.

Answer 1

Let the number of sides of a polygon be n.

The smallest angle = 120° = a

Common difference in angles = d = 5°

Now, in a polygon of n sides, the sum of interior angles = (2n – 4) × 90°

⇒ `n/2 [2 xx 120^circ + (n - 1) × 5^circ] = (2n - 4) xx 90^circ`

⇒ `n/2 [240^circ + 5n - 5^circ] = 180n - 360^circ`

⇒ n[235° + 5n] = 360n – 720°

⇒ 5n² – 125n + 720 = 0

⇒ n² – 25n + 144 = 0

⇒ n² – 16n – 9n + 144 = 0

⇒ n(n – 16) – 9(n – 16) = 0

⇒ (n – 16)(n – 9) = 0

⇒ n = 16 or n = 9