Question

The interior angles of the polygon are in A.P. If the smallest angle is ${100^0}$ and the common difference is ${4^0}$, find the number of sides ?

Answer 1

Hint: In this question apply the property that the sum of all interior angles in the polygon is ${S_n} = \left( {n - 2} \right) \times {180^0}$ later on apply the formula of sum of an A.P so use these concepts to reach the solution of the question.

Complete step-by-step answer:

As we know that the sum of all interior angles in the polygon is ${S_n} = \left( {n - 2} \right) \times {180^0}$ where n is the number of sides.
Now it is given that the interior angles are in A.P.
And the smallest angle $\left( {{a_1}} \right) = {100^0}$.
And the common difference (d) = ${4^0}$.
Then we have to find out the number of sides.
Now as we know that the sum of an A.P is given as
${S_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right)$ so, use this formula to calculate the number of sides.
So, substitute all these values in this equation we have,
$ \Rightarrow \left( {n - 2} \right){180^0} = \dfrac{n}{2}\left( {2 \times {{100}^0} + \left( {n - 1} \right){4^0}} \right)$
Now simplify this equation we have,
$ \Rightarrow 180n - 360 = n\left( {100 + 2n - 2} \right)$
$ \Rightarrow 2{n^2} + 98n - 180n + 360 = 0$
$ \Rightarrow 2{n^2} - 82n + 360 = 0$
Divide by 2 throughout we have,
$ \Rightarrow {n^2} - 41n + 180 = 0$
Now factorize this equation we have,
$ \Rightarrow {n^2} - 5n - 36n + 180 = 0$
$ \Rightarrow n\left( {n - 5} \right) - 36\left( {n - 5} \right) = 0$
$ \Rightarrow \left( {n - 5} \right)\left( {n - 36} \right) = 0$
$ \Rightarrow n = 5,36$
Now as we know that the last term of an A.P is calculated as
$ \Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d$
So for n = 36 we have
$ \Rightarrow {a_{36}} = {100^0} + \left( {36 - 1} \right){4^0} = {100^0} + {140^0} = {240^0}$
So, this is not possible as the biggest angle cannot be greater than ${180^0}$.
Now, for n = 5 we have
$ \Rightarrow {a_5} = {100^0} + \left( {5 - 1} \right){4^0} = {100^0} + {16^0} = {116^0}$
This is possible.
So, the number of sides of the polygon are 5.
So, this is the required answer.

Note: In such types of questions the key concept we have to remember is that the sum of all the interior angles of the polygon is ${S_n} = \left( {n - 2} \right) \times {180^0}$ and always recall the formula of sum of an A.P then substitute all the values in the formula and simplify, then check whether the highest angle of the polygon is greater than 180 degree or not if yes then this solution is not the valid answer and if not then the solution is valid answer as the highest interior angle of the polygon is always less than 180 degree.