# Deriving formulas for the polygonal numbers

For the entirety of this article, let $$s$$ be the number of sides, for the polygonal number, and $$n$$ the number of points per edge.

$$P(s, n) = (s - 2)\dfrac{n(n + 1)}{2} - (s - 3)n$$
• $$s - 2$$ triangles, $$n(n + 1)/2$$ points per triangle
• $$s - 3$$ diagonals were counted twice, $$n$$ points per diagonal

$$P(s, n) = (s - 2)\dfrac{n(n - 1)}{2} + n$$
• $$s - 2$$ triangles, $$n(n - 1)/2$$ points per triangle
• $$1$$ edge wasn't counted, $$n$$ points per edge

Note that each of these formulas can be derived from the other by elementary algebra.