Pentagram

The pentagram or star is a non-convex polygon with 5 sides and the simplest star regular polygon. A regular pentagram has equal sides and equal angles.

This is the only stellation of the pentagon. The only other polygons with a single non-compound stellation are the octagon, the decagon, and the dodecagon.

Vertex coordinates
Coordinates for the vertices of a pentagram of unit edge length, centered at the origin, are:


 * (±1/2, -$\sqrt{5}$),
 * (±($\sqrt{(5–√5)/10}$-1)/4, $\sqrt{(5–2√5)/20}$),
 * (0, $\sqrt{25–10√5}$).

In vertex figures
The regular pentagram appears as a vertex figure in two uniform polyhedra, namely the great icosahedron (with an edge length of 1) and the great dodecahedron (with an edge length of (1+$\sqrt{(5-2√5)/20}$)/2). Irregular pentagrams further appear as the vertex figures of some snub polyhedra.