Question

How many sides does a regular polygon have if each of its interior angles is $172.8$ degrees?

Answer 1

As we know that a regular polygon is a polygon that is equiangular and equilateral which means it has equal sides and equal angles. We know that in a regular polygon if the interior angle is $x$ then the exterior angle is $(180 - x)$ because it forms a linear pair. These are two dimensional figures.

Complete step by step solution:
As per the question we have an interior angle i.e. $x = {172.8^ \circ }$. So the value of exterior angle is $(180 - x) = (180 - 172.8)$, It gives us the value ${7.2^ \circ }$.
Also we know that the sum of all exterior angles of any polygon whether it is regular or not is $360$ degrees.
Therefore the number of sides is $\dfrac{{360}}{{angle}}$, here angle means each exterior angle.
So by putting the value we have $\dfrac{{360}}{{7.2}} = 50$angles. As we know that a regular polygon has an equal number of sides and angles so the number of angles is equal to the number of sides i.e. 50 sides.
Hence the number of sides a regular polygon has if its interior angle is $172.8$ degrees are $50$.

Note: We know that the sum of all exterior angles in any polygon is $360$degrees. By dividing the degree from each exterior angle gives the number of sides it can have. So we first found the value of the exterior angle from the given interior angle value by applying the linear pair formula. And then after that we can easily calculate the number of sides.