Question
Answers

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

Answer Verified Verified
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.

Complete step-by-step answer:
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.