Question

Each interior angle of the regular n-gon has a measure of $165{}^\circ $ . How many sides does n-gon have?

Answer 1

Hint: Here, We know that a polygon is named on the basis of its sides and in every regular polygon that has $n$ sides, generates $\left( n-2 \right)$ regular equilateral triangles. Since, the sum of the interior angles of a triangle is $180{}^\circ $ . Then:
$\Rightarrow \text{Number of triangle in a regular polygon}\times \text{sum of the interior angles of a triangle }$
This is equal to
$\Rightarrow \text{Sum of interior angles of a regular polygon}$

Complete step-by-step answer:
In the question, there is a given regular polygon that is named n-gon that means this regular polygon has $n$ sides.
$\Rightarrow \text{No}.\text{ of sides in the given regular Polygon}=n$
Since, we know that every polygon has some numbers of equilateral triangles that is number of sides minus two as:
\[\Rightarrow \text{No}.\text{ of equilateral triangles }=\text{ number of sides in the given regular polygon }-\text{ 2}\]
\[\Rightarrow \text{No}.\text{ of equilateral triangles }=\text{ }\left( \text{n}-\text{2} \right)\]
We all know that sum of the interior angles of an equilateral triangle is $180{}^\circ $ . So, we can write it as:
$\Rightarrow \text{The sum of the interior angles of an equilateral triangle }=\text{ 18}0{}^\circ $
From the previous step we know that the sum of interior angles of an equilateral triangle is $180{}^\circ $ . Since, the given regular polygon has \[\left( \text{n}-\text{2} \right)\] equilateral triangle, the sum of interior angles of \[\left( \text{n}-\text{2} \right)\]equilateral triangle is equal to:
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }\left( n-2 \right)\times 180{}^\circ \]
After opening the bracket the above equation can be as:
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }\left( n\times 180{}^\circ -2\times 180{}^\circ \right)\]
After doing necessary calculation, the above equation will be as:
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }180{}^\circ n-360{}^\circ \] … $\left( i \right)$
Since, each interior angle of the regular polygon is given. Therefore, the sum of the interior angles of the given regular polygon is equal to:
\[\begin{align}
  & \Rightarrow \text{The sum of the interior angles of a regular Polygon = } \\
 & \text{Each interior angle of the regular Polygon }\times \text{ number of sides in the regular Polygon} \\
\end{align}\]
Since, we have angle of each interior angle of the given regular n-gon $165{}^\circ $ . So, we will see that the above equation is also describes the sum of the interior angles of the given polygon as:
\[\Rightarrow \text{The sum of the interior angles of a regular Polygon = 165}{}^\circ \text{ }\times \text{ n}\]
The above equation can be written as:
\[\Rightarrow \text{The sum of the interior angles of a regular Polygon = 165}{}^\circ \text{ n}\] … $\left( ii \right)$
Since, both the equation $\left( i \right)$ and equation $\left( ii \right)$describes the sum of the interior angles of the given polygon, we can see that :
$\Rightarrow \text{The sum of interior angles of a regular polygon = The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle }$Using the equation $\left( i \right)$ and equation $\left( ii \right)$ we can write the above equation as:
\[\Rightarrow \text{165}{}^\circ \text{ n= }180{}^\circ n-360{}^\circ \]
Now, we will combine equal like terms as:
\[\Rightarrow 180{}^\circ n-\text{165}{}^\circ \text{ n}=360{}^\circ \]
After subtracting \[\text{165}{}^\circ \text{ n}\] from \[\text{180}{}^\circ \text{ n}\] we will get \[\text{15}{}^\circ \text{ n}\] as:
\[\Rightarrow 1\text{5}{}^\circ \text{ n}=360{}^\circ \]
\[\Rightarrow \text{ n}=\dfrac{360{}^\circ }{1\text{5}{}^\circ }\]
\[\Rightarrow \text{ n}=24\]
Hence, the number of sides in the given polygon is \[24\] .

Note: Here, we can check that the obtaining value is right or not by putting the values in equation one and equation two. If the both values are same that means the obtaining number of sides is correct.
So, we will start from equation $\left( i \right)$ :
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }180{}^\circ n-360{}^\circ \]
Here, we will put \[\text{ n}=24\]. So the above equation will be:
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }180{}^\circ \times 24-360{}^\circ \]
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = 4320}{}^\circ -360{}^\circ \]
\[\Rightarrow \text{The sum of interior angles of }\left( \text{n}-\text{2} \right)\text{equilateral triangle = }3960{}^\circ \]
Now, we will use equation $\left( ii \right)$:
\[\Rightarrow \text{The sum of the interior angles of a regular Polygon = 165}{}^\circ \text{ n}\]
Here, we will put \[\text{ n}=24\] , so that the above equation will be:
\[\Rightarrow \text{The sum of the interior angles of a regular Polygon = 165}{}^\circ \text{ }\times \text{24}\]
\[\Rightarrow \text{The sum of the interior angles of a regular Polygon = 3960}{}^\circ \]
Since, we got the same value from both the equations. Hence, the solution is correct.