Question

The interior angles of a polygon are in AP. The smallest angle is 52\[^\circ \]and the common difference is 8\[^\circ \]. Find the number of sides of the polygon.

Answer 1

Hint: We are given the first term of the AP, common difference is also provided so the rest of the series can be formed easily. We need to find the no. of sides which cannot be found directly. So, we use the formula for the sum of the polygon of n side and equate it to the sum upto nth term of the AP and proceed.

Complete step by step answer:
First we will form the AP, d = 8
52\[^\circ \], 62\[^\circ \],68\[^\circ \] and so on.
Let the number of sides be n.
Sum of polygon of n sides is given by = (n-2)×180\[^\circ \]
∴ \[{S_n} = 180^\circ - 360^\circ \]……….(1)
But he sum of n term of AP is given by
\[{S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]\]……….(2), where a is the first term of the AP
Equating (1) and (2), we get
\[ \Rightarrow 180n - 360 = \dfrac{n}{2}\left[ {2 \times 52 + (n - 1)} \right]\]
\[ \Rightarrow 180n - 360 = \dfrac{n}{2}(104 - 8n - 8)\]
\[ \Rightarrow 180n - 360 = \dfrac{n}{2}(96 + 8n)\]
\[ \Rightarrow 180n - 360 = 48n + 4{n^2}\]
\[ \Rightarrow 4{n^2} - 132n + 360 = 0\]
\[ \Rightarrow {n^2} - 33n + 90 = 0\]
\[ \Rightarrow {n^2} - 3n - 30n + 90 = 0\]
\[ \Rightarrow n(n - 3) - 30(n - 3) = 0\]
\[ \Rightarrow (n - 3)(n - 30) = 0\]
∴ n = 3 or 30
To find out the exact value of n we’ll use the nth term formula
\[{a_n} = a + (n - 1)d\]
\[{a_n}\]=52+29×8
\[{a_n}\]=52+232
\[{a_n}\]=284, this is a reflexive angle and we are given the interior angle
∴ n=30 is never possible.
∴ n=3 is the right answer.

Note: A polygon is a two dimensional shape formed with the straight lines. Triangles, quadrilaterals, pentagons and hexagons are all examples of polygons. The name tells us how many sides the shape has.