Question

Prove that the interior angle of a regular five sided polygon (Pentagon) is three times the exterior angle of a regular decagon.

Answer 1

Hint: In this question, we have to prove the interior angle of regular Pentagon to be equal to three times the exterior angle of regular decagon. For this, we will find values of each interior angle of Pentagon and each exterior angle of decagon. After that, we will compare it to prove our result. For finding interior angle of any polygon, we use formula: $\text{Each angle}=\dfrac{\left( n-2 \right)}{n}\times {{180}^{\circ }}$
Where n is number of side of the polygon. For finding exterior angle of any polygon, we use formula: $\text{Exterior angle}=\dfrac{{{360}^{\circ }}}{n}$
Where n is the number of sides of polygon.

Complete step by step answer:
Here we have to prove that the interior angle of a regular Pentagon is three times the exterior angle of a regular decagon.
A regular Pentagon is a five sided polygon with each side equal to each other. Let us find the value of each interior angle of a regular Pentagon.
As we know, formula for finding value of interior angle of regular polygon is given by $\dfrac{\left( n-2 \right)}{n}\times {{180}^{\circ }}$ where n is number of sides of polygon. So, we will use this formula for finding the interior angle of the Pentagon. Since, the number of sides in the Pentagon are 5. Therefore, n = 5. Hence,
\[\begin{align}
  & \text{Interior angle}=\dfrac{\left( 5-2 \right)}{5}\times {{180}^{\circ }} \\
 & \Rightarrow \dfrac{3}{5}\times {{180}^{\circ }} \\
 & \Rightarrow {{108}^{\circ }} \\
\end{align}\]
Each interior angle of a regular Pentagon is ${{108}^{\circ }}$.
Now, let us find the exterior angle of a regular decagon.
As we know, the formula for finding the value of the exterior angle of a regular polygon is given by $\dfrac{{{360}^{\circ }}}{n}$ where n is the number of sides of a polygon. So, we will use this formula for finding the exterior angle of a decagon. Since, the number of sides of a decagon is 10. Therefore, n = 10. Hence,
\[\text{Exterior angle}=\dfrac{{{360}^{\circ }}}{10}={{36}^{\circ }}\]
Each exterior angle of regular decagon is ${{36}^{\circ }}$.
Now, interior angle of regular Pentagon can be written as ${{108}^{\circ }}=3\times {{36}^{\circ }}$ and ${{36}^{\circ }}$ is value of each exterior angle of regular decagon. Therefore, the interior angle of a regular Pentagon is 3 times the exterior angle of a regular decagon.

Note: Students should note that formulas for finding interior angle and exterior angle of polygon are applicable only on regular polygons. Since, sum of exterior angles of any polygon is ${{360}^{\circ }}$ therefore, each angle will be equal to $\dfrac{{{360}^{\circ }}}{n}$ where n will be number of sides or number of angles. Students should know notation of polygons with different numbers of sides.