Question

# The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of sides of the polygon.

Hint: Here we will use the arithmetic progressions to calculate the number of sides of the polygon. We are given the difference between the two consecutive angles and it will be the common difference and the smallest angle provided will be the first term of the A. P. We will use the formula of sum of the n terms in an A. P. ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$. We will then compare the obtained sum with the sum obtained by the formula: Sum of all angles of a polygon with n sides = ${180^ \circ }(n - 2)$. Upon comparing, we will solve the obtained equation for the values of n.

We are given the difference between two consecutive angles of a polygon as 5°.
The smallest angle of the given polygon = 120°
So, according to the question, the second smallest angle = 120° + 5° = 125°
Similarly, the third angle will be 125° + 5°= 130°
Therefore, the angles form an A. P. as 120°, 125°, 130°, …
In this A. P., the first term: a = 120°
The common difference: d = 5°
So, we can use the sum of n terms of an A. P. given by ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] \\ \Rightarrow {S_n} = \dfrac{n}{2}\left[ {\left( {2 \times {{120}^ \circ }} \right) + \left( {n - 1} \right){5^ \circ }} \right] \\$
Now, we know that the sum of all angles of a polygon is given by: SP = ${180^ \circ }(n - 2)$
Both of the equations must be equal because the sum of all the angles of a polygon is definite irrespective of the method used.
$\Rightarrow {S_n} = {S_p} \\ \Rightarrow \dfrac{n}{2}\left[ {2\left( {120} \right) + \left( {n - 1} \right)5} \right] = 180\left( {n - 2} \right) \\$
$\Rightarrow n\left[ {240 + 5n - 5} \right] = 360\left( {n - 2} \right) \\ \Rightarrow 240n + 5{n^2} - 5n = 360n - 720 \\ \Rightarrow 5{n^2} - 125n + 720 = 0 \\$
Now, we will solve this quadratic equation for the values of n.
$\Rightarrow 5\left( {{n^2} - 25 + 144} \right) = 0 \\ \Rightarrow {n^2} - 25n + 144 = 0 \\ \Rightarrow {n^2} - 16n - 9n + 144 = 0 \\ \Rightarrow n\left( {n - 16} \right) - 9\left( {n - 16} \right) = 0 \\ \Rightarrow \left( {n - 9} \right)\left( {n - 16} \right) = 0 \\ \Rightarrow n = 9or16 \\$
We get two values of n and two different numbers of sides in a polygon can’t be acceptable.
So, let us check for the value of n.
For n = 16, $a_{16}$ = a + 15d = 120 + 75 = 195 which is not possible as 195 > 180.
Hence, the total number of sides in the given polygon are 9.

Note: Such questions are confusing as they require quite a number of formulae. You may get confused after getting the sum of n terms using the A. P. in how to proceed further. Always make sure to check for the obtained values because there might be a possibility that any one can not satisfy the conditions.