Question

The interior angles of a polygon are in AP. The smallest angle is 52°, and the common difference is 8°. Find the number of sides of the polygon.

Answer 1

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 = n/2{2a+(n-1)d}

Sum of angles of the given polygon is n/2{(2x52°)+(n-1)x8°}.

Hint:

Sum of interior angles of a polygon of n sides is (n-2)x8°

Therefore,

(n-2)x180° = n/2{104°+(n-1)x8°}

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.