Question

Interior angles of a polygon are in A.P. Smallest interior angle is$52^\circ $and the difference of consecutive interior angles is$8^\circ $, then find the number of sides of the polygon.

Answer 1

Hint: From the given question we know that the angles of polygon are in AP that is, Sum to n number ${S_n} = \dfrac{n}{2}\left( {2\left( a \right) + \left( {n - 1} \right)d} \right)$should be equal to sum of measures of the interior angles of polygon with n sides is$\left( {n - 2} \right)180$. Here the smallest angle ‘a’ and difference between angles given is ‘d’ are given in the equation. This forms an equation to solve to get the number of sides of a polygon.

Complete step-by-step answer:
Given, smallest interior angle, a=$52^\circ $
And difference of consecutive interior angles, d=$8^\circ $
Interior Angle: An interior angle of a polygon is an angle inside the polygon at one of its vertices.
Sum to n numbers is given by${S_n} = \dfrac{n}{2}\left( {2\left( a \right) + \left( {n - 1} \right)d} \right)$
Here ‘a’ is the smallest angle and d is the difference of consecutive interior angle (common ratio).
That is, ${S_n} = \dfrac{n}{2}\left( {2\left( {52} \right) + \left( {n - 1} \right)8} \right)$
$ \Rightarrow \dfrac{n}{2}\left( {104 + 8n - 8} \right)$
$ \Rightarrow \dfrac{n}{2}\left( {96 + 8n} \right)$… (1)
And we know that sum of measures of the interior angles of polygon with n sides is$\left( {n - 2} \right)180$.
Therefore,$\left( {n - 2} \right)180$… (2).
As the equation (1) and (2) are the same,
Now equate (1) and (2)
$\dfrac{n}{2}\left( {96 + 8n} \right) = \left( {n - 2} \right)180$
$ \Rightarrow n\left( {96 + 8n} \right) = \left( {n - 2} \right)180 \times 2$
$ \Rightarrow 8{n^2} + 96n = 360n - 720$
$ \Rightarrow 8{n^2} + 96n - 360n + 720 = 0$
$ \Rightarrow 8{n^2} - 264n + 720 = 0$
Divide the complete equation by 8
$ \Rightarrow {n^2} - 33n + 90 = 0$
$ \Rightarrow {n^2} - 30n - 3n + 90 = 0$
$ \Rightarrow n\left( {n - 30} \right) - 3\left( {n - 30} \right) = 0$
$ \Rightarrow \left( {n - 30} \right)\left( {n - 3} \right) = 0$
Therefore $n = 3$and$n = 30$
As value of n can’t be 30 as the 30th angle will be greater than$180^\circ $, so$n = 3$
Therefore, the number of sides in a polygon is 3.

Note: The measure of each interior angle of an equiangular polygon is$\dfrac{{\left( {n - 2} \right)180}}{n}{\text{ }}or{\text{ }}180 - \dfrac{{360}}{n}$ (the supplement of an exterior angle). If you count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always$360^\circ $. The measure of each exterior angle of an equiangular polygon is$\dfrac{{360}}{n}$.