Question

# The number of sides of a regular polygon, if each of its interior angles is 135 degrees, is given by _____(a) 4(b) 6(c) 8(d) 10

Hint: We are given a regular polygon with each interior angle as ${{135}^{\circ }}.$ We have to find the number of sides by using the relation between the number of sides and the interior angle. We will consider the polygon has n sides. Then the relation between the interior angle and the side is
$\text{Sum of all interior angles}=\left( n-2 \right){{180}^{\circ }}$
So, we will get,
$\text{n}\times \text{13}{{\text{5}}^{\circ }}=\left( n-2 \right){{180}^{\circ }}$
We will solve for n and then get the number of sides of the polygon.

We are given that each interior angle is ${{135}^{\circ }},$ we have to find the number of sides of the polygon. Let’s start with the assumption that the number of sides be n, this means that we have n sided regular polygon whose interior angle is ${{135}^{\circ }}.$
We know that their relation between the interior angle and the number of sides of the polygon is given as,
$\text{Sum of all interior angles}=\left( n-2 \right)\times {{180}^{\circ }}$
Our n sided regular polygon has n interior angles with each of 135 degrees. So, the sum of all the interior angles will be
$\text{Sum of all interior angles}=n\times {{135}^{\circ }}$
So, as we have the sum of all the interior angles given as $\left( n-2 \right)\times {{180}^{\circ }},$ so, we get,
$\Rightarrow n\times {{135}^{\circ }}=\left( n-2 \right)\times {{180}^{\circ }}$
Now, opening the brackets, we get,
$\Rightarrow 135n=180n-{{360}^{\circ }}$
Now simplifying the term, we get,
$\Rightarrow 136n-180n=-360$
Solving for n, we get,
$\Rightarrow -45n=-360$
Dividing both sides by – 45, we get,
$n=\dfrac{-360}{-45}=8$
Therefore, we get the number of sides that the polygon has as 8.
So, the correct answer is “Option C”.

Note: We can cross-check why other options are not correct. We know that the interior angle and the sides are related and this relation is given as
$\text{n}\times \text{Interior Angles}=\left( n-2 \right)\times {{180}^{\circ }}$
As the interior angle is ${{135}^{\circ }},$ so we get,
$n\times {{135}^{\circ }}=\left( n-2 \right){{180}^{\circ }}$
(a) If we take n = 4,
$n\times {{135}^{\circ }}=4\times {{135}^{\circ }}={{540}^{\circ }}$
While,
$\left( n-2 \right){{180}^{\circ }}=\left( 4-2 \right){{180}^{\circ }}={{360}^{\circ }}$
Both are not equal and hence (a) is not the right option.
(b) If we take n = 6,
$n\times {{135}^{\circ }}=6\times {{135}^{\circ }}={{810}^{\circ }}$
While,
$\left( n-2 \right){{180}^{\circ }}=\left( 6-2 \right){{180}^{\circ }}={{720}^{\circ }}$
Both are not equal and hence (b) is not the right option.
(c) If we take n = 10,
$n\times {{135}^{\circ }}=10\times {{135}^{\circ }}={{1350}^{\circ }}$
While,
$\left( n-2 \right){{180}^{\circ }}=\left( 10-2 \right){{180}^{\circ }}={{1440}^{\circ }}$
Both are not equal and hence (c) is not the right option.