Question

Is it possible to have a regular polygon whose each interior angle is
(i) ${{170}^{\circ }}$ (ii) ${{138}^{\circ }}$

Answer 1

Hint: So, we first describe how the interior and exterior angles of a n-sided regular polygon works. We find their general values. Then using the given values of interior angle, we find the value of exterior angles. We use that to find the number of sides of the polygon from the formula of exterior angles $\dfrac{2\pi }{n}$. If the value is an integer then the polygon exists and if the value is a fraction then the polygon doesn’t exist.

Complete step by step answer:
We know that for an n-sided regular polygon, the exterior angles would be all equal and the value will be $\dfrac{2\pi }{n}$.
Now from the given values of interior angles, we found the exterior angles as the sum of interior and exterior angles is $\pi $. We use that value to find if the value of n is integer or fraction. If it’s an integer then the polygon exists and if it’s a fraction then the polygon doesn’t exist.
For the interior angle ${{170}^{\circ }}$. The exterior angle would be $\pi -{{170}^{\circ }}={{180}^{\circ }}-{{170}^{\circ }}={{10}^{\circ }}$.
Now we assume the polygon is k-sided then value of the exterior angles will be $\dfrac{2\pi }{k}$ which will be equal to ${{10}^{\circ }}$.
So, $\dfrac{2\pi }{k}={{10}^{\circ }}$. We solve it to get value of k as $k=\dfrac{{{360}^{\circ }}}{{{10}^{\circ }}}=36$. It’s an integer value.
The regular polygon exists and is of 36 sides.
For the interior angle ${{138}^{\circ }}$. The exterior angle would be $\pi -{{138}^{\circ }}={{180}^{\circ }}-{{138}^{\circ }}={{42}^{\circ }}$.
Now we assume the polygon is p-sided then value of the exterior angles will be $\dfrac{2\pi }{p}$ which will be equal to ${{42}^{\circ }}$.
So, $\dfrac{2\pi }{p}={{42}^{\circ }}$. We solve it to get value of k as \[p=\dfrac{{{360}^{\circ }}}{{{42}^{\circ }}}=\dfrac{60}{7}\]. It’s not an integer value.
The regular polygon doesn’t exist.

Note:
 We also can use the formula of interior angles to find the values. We know that for an n-sided regular polygon, the interior angles would be all equal and the value will be $\dfrac{\pi }{n}\left( n-2 \right)$. We put the values of the given interior angles we try to find the value of n. if they are integer then polygon exists and if the value is fraction then the polygon doesn’t exist.