The interior angles of a polygon are in AP. The smallest angle is 520, and the common difference is 80. Find the number of sides of the polygon.
Given:
Interior angles of a polygon are in A.P
Smallest angle = a = 52°
Common difference = d = 8°
Let the number of sides of a polygon = n
Angles are in the following order
52°, 52° + d, 52° + 2d, ........, 52° + (n - 1) ×d
Sum of n terms in A.P = s
Sum of angles of the given polygon is
Hint:
Sum of interior angles of a polygon of n sides is
Therefore,
180n - 360 = 52n + n (n - 1) ×4
4n2 + 48n = 180n - 360
4n2 - 132n + 360 = 0
n2 - 33n + 90 = 0
(n - 3)(n - 30) = 0
n = 3 &n = 30
∴ It can be a triangle or a 30 sided polygon.
The number of sides of the polygon is 3 or 30.