Let us consider any regular n-gon. It must follow that all of the angles are congruent. Similarly, all of the angles must have an equal number of diagonals before any are constructed.

Let us consider an arbitrary edge: it will have n - 3 diagonals that can be constructed, since the sides that form its vertex and its own cannot be drawn. It follows that the next vertex will also have n - 3 diagonals capable of construction (because it could be regarded as the side of the previous vertex). If one continues to construct the diagonals, the remaining vertices will have n - 4, n - 5,... 1 vertices remaining (because the vertices cannot be regarded as the sides that formed the previous vertex considering all vertices have a set of sides that are unique).

Ergo, the series is as follows:

We can use the above formula to find the diagonals of any regular polygon.