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How many sides does a regular polygon have if each of its exterior angle measures ${{40}^{\circ }}$?

Last updated date: 22nd Feb 2024
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IVSAT 2024
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Hint: 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 exterior angle and the formula of exterior angles $\dfrac{2\pi }{n}$, we find the number of sides. If the value is integer, then polygon exists and if value is fraction, then the polygon doesn’t exist.

Complete step-by-step solution:
We know that for a 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 integer then the polygon exists and if it’s a fraction then the polygon doesn’t exist.
For the exterior angle ${{40}^{\circ }}$, we assume the polygon is p-sided then the value of the exterior angles will be $\dfrac{2\pi }{p}$.
So, $\dfrac{2\pi }{p}=40$. We solve it to get the value of p as \[p=\dfrac{360}{40}=9\].
The regular polygon exists and has 9 sides.

Note: We also can use the formula of interior angles to find the values. We know that for a n-sided regular polygon, the interior angles would all be equal and the value will be $\dfrac{\pi }{n}\left( n-2 \right)$. We put the values of the given interior angles $180-40-140$ and try to find the value of n.