Q. 484.3( 8 Votes )

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 :

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. Sn = 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.


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