# Polygonal Number

## Polygonal Number

Figurate number that we can represent by a convex regular polygon or by a sequence that is partially superimposed on convex regular polygons.

If the convex regular polygon has $$c$$ sides and if, on each side, there are $$n$$ points arranged, vertices included, then we distinguish two kinds of polygonal numbers:

1. Those that are represented on the perimeter of the polygon: $$\textrm{p}_{n}^{c}$$ = $$c$$($$n$$ – 1)
2. Those that are represented on the closed surface of the polygon: $$\textrm{P}_{n}^{c}$$ = $$\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}$$

### Example

1. $$\textrm{p}_{n}^{3}$$ = 3($$n$$ – 1)   et   $$\textrm{p}_{5}^{3}$$ = 3(5 – 1) = 12
2. $$\textrm{P}_{n}^{c}$$ = $$\dfrac{(c\space –\space 2)n^2 –\space (c\space –\space 4)n}{2}$$   et   $$\textrm{P}_{n}^{3}$$ = $$\dfrac{n(n + 1)}{2}$$  et  $$\textrm{P}_{5}^{3}$$ = $$\dfrac{5(5 + 1)}{2}$$ = 15