Number of sides of a regular polygon

The sum of interior angles of a polygon is $S=(n-2)(180)$ . The sum of exterior angles of a polygon is $360$ . Let $n$ be the number of sides of the regular polygon, $\theta$ be the measure of one interior angle of the regular polygon and $\beta$ be the measure of one exterior angle of the regular polygon.

Then,

$\theta=4\beta$

$\dfrac{S}{n}=4\dfrac{360}{n}$

$\dfrac{(n-2)(180)}{n}=\dfrac{1440}{n}$

$n-2=\dfrac{1440}{180}$

$n-2=8$

$n=10$

Let x = exterior angle. Then 180 - x = interior angle.

180 - x = 4x

180 = 5x

36 = x

360 / 36 = 10 sides