Sides from the Exterior

The exterior angle gives the number of sides of a regular polygon quickly. Each exterior angle is $\frac{360}{n}$, so if every exterior angle is 45°, then $\frac{360}{n} = 45$.

Multiply both sides by n to bring it out of the bottom: $360 = 45\cdot n$. Divide by 45: $n = 8$. The polygon is a regular octagon.

From an interior angle, take it from 180° first. A regular polygon with 135° in each corner has exterior angles of 45°, so it has 8 sides, with no equation to expand.