Question

Each interior angle of a regular convex polygon measures $162{}^\circ $ . How many sides does the polygon have?

Answer 1

Hint: To get the sides of a regular polygon, we will follow some principles of polygon as:
The number of sides of a regular polygon is denoted by a symbol. Here, we will use $n$ .
The each exterior angle of a regular polygon \[=180{}^\circ -\text{Each interior angle of a regular polygon}\]
The sum of the exterior angles of a regular polygon is always equal to $360{}^\circ $ and the formula for getting each exterior angle of a regular polygon is equal to the value obtained from $360{}^\circ $divided by $n$.

Complete step by step solution:
Since, in the question we have a regular convex polygon.
So let us consider that the number of sides of the given regular convex polygon is $n$ .
Here, we will start by getting the exterior angle of the given regular convex polygon. Since, the sum of the exterior and interior angles of a regular polygon is equal to $180{}^\circ $ . So, we can get the value of exterior angle by using this method as:
\[\Rightarrow \text{Each exterior angle of a regular polygon }+\text{ Each interior angle of a regular polygon }=\text{ 18}0{}^\circ \]
Since, we have the value of each interior angle from the question. We will use that value as:
\[\Rightarrow \text{Each exterior angle of a regular polygon }+\text{ 162}{}^\circ \text{ }=\text{ 18}0{}^\circ \]
Now, we will place all the degrees one side as:
\[\Rightarrow \text{Each exterior angle of a regular polygon }=\text{ 18}0{}^\circ -\text{ 162}{}^\circ \]
After subtracting the above values, we will get:
\[\Rightarrow \text{Each exterior angle of a regular polygon }=\text{ 18}{}^\circ \]
Here, we got the each exterior angle of the given regular polygon as $18{}^\circ $ .
Since, we know that the sum of the exterior angles of a regular polygon is $360{}^\circ $and we have some value as the number of sides is $n$ and each exterior angle is $18{}^\circ $ . We will use the formula as:
\[\Rightarrow \text{Sum of the exterior angle of a regular polygon }=\text{ number of sides }\times \text{ Each exterior angle of regular polygon}\]
Since, we already have some values; after applying them in the above equation, the above equation will be as:
\[\Rightarrow \text{36}0{}^\circ =\text{ n }\times 18{}^\circ \text{ }\]
Now, we will do necessary calculations as:
\[\Rightarrow n=\text{ }\dfrac{\text{36}0{}^\circ }{18{}^\circ }\]
\[\Rightarrow n=\text{ 20}\]
Hence, the number of sides is \[\text{20}\] .

Note: Here, we will verify that the obtained number of sides is correct or not by using the following formula as:
The formula for sum of the exterior angles is:
$\Rightarrow n\times \text{ each interior angle of a regular polygon = }\left( n-2 \right)\times 180{}^\circ $
Now, we will use \[n=\text{ 20}\] in the above equation, the equation will be as:
$\Rightarrow 20\times \text{ 162}{}^\circ \text{ = }\left( 20-2 \right)\times 180{}^\circ $
After multiplication, we will have:
$\Rightarrow 3240{}^\circ \text{ = 18}\times 180{}^\circ $
$\Rightarrow 3240{}^\circ \text{ = 324}0{}^\circ $
Therefore, \[\text{L}.\text{H}.\text{S}.\text{ }=\text{ R}.\text{H}.\text{S}.\]
Hence, the solution is correct for the given question.