From an Angle to the Sides

In a regular polygon with n sides, the angle sum (n - 2) × 180 is shared equally between n corners. To go backwards, from one angle to the number of sides, write the angle sum two ways and set them equal.

Say every angle is 150°. The n angles total 150 × n, and the angle sum is also (n - 2) × 180. So (n - 2) × 180 = 150 × n. Multiply out the bracket: 180 × n - 360 = 150 × n. Take 150n from both sides: 30 × n = 360, so n = 12. A regular polygon with 12 sides.

If n does not come out whole, no regular polygon has that angle.