Question

The interior angles of a polygon are in A.P., the smallest angle is 75° and the common difference is 10°. Find the number of sides of the polygon.

Answer 1

Given: The smallest angle is 75°

i.e. a = 75

and common difference = 10°

i.e. d = 10

Therefore, the series is

75, 85, 95, 105, …

and the sum of interior angles of a polygon =(n – 2) 180°

i.e. S_n = 180

We know that,

⇒ (n – 2)360 = n [140+10n]

⇒ 360n – 720 = 140n + 10n²

⇒ 36n – 72 – 14n – n² = 0

⇒ n² – 22n + 72 = 0

⇒ n² – 18n – 4n + 72 = 0

⇒ n(n – 18) – 4(n – 18) = 0

⇒ (n – 4)(n – 18) = 0

Putting both the factor equal to 0, we get

n – 4 = 0 or n – 18 = 0

⇒ n = 4 or n = 18

Hence, the number of sides of a polygon can be 4 or 18.