Question

The interior angles of a polygon are in arithmetic progression.

The smallest angle is 120^•and the common difference is 5.

Find ND the number of sides of polygon.

Answer 1

Let there be in n sides in the polygon.
Then by geometry, sum of all n interior angles of polygon = (n – 2) * 180°

Also the angles are in A. P. with the smallest angle = 120° , common difference = 5°

∴ Sum of all interior angles of polygon
= n/2[2 * 120 + ( n – 1) * 5

Thus we should have
n/2 [2 * 120 + (n – 1) * 5] = (n – 2) * 180
⇒ n/2 [5n + 235] = (n – 2 ) * 180
⇒ 5n² + 235n = 360n – 720
⇒ 5n² – 125n + 720 = 0 ⇒ n² – 25n + 144 = 0
⇒ (n – 16 ) (n – 9) = 0 ⇒ n = 16, 9

Also if n = 16 then 16^th angle = 120 + 15 * 5 = 195° > 180°
∴ not possible.
Hence n = 9.