# Pentagram

Pentagram
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymStar
Coxeter diagramx5/2o ()
Schläfli symbol{5/2}
Elements
Edges5
Vertices5
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573}$
Inradius${\displaystyle \sqrt{\frac{5-2\sqrt5}{20}} ≈ 0.16246}$
Area${\displaystyle \frac{\sqrt{25-10\sqrt5}}{4} ≈ 0.40615}$
Angle36°
Central density2
Number of pieces10
Level of complexity2
Related polytopes
ArmyPeg
DualPentagram
ConjugatePentagon
Convex corePentagon
Abstract properties
Flag count10
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH2, order 10
ConvexNo
NatureTame

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.

Pentagrams occur as faces in two of the four Kepler-Poinsot solids, namely the small stellated dodecahedron and great stellated dodecahedron.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,\sqrt{\frac{5+\sqrt5}{40}}\right),}$
• ${\displaystyle \left(0,\,-\sqrt{\frac{5-\sqrt5}{10}}\right).}$

## Representations

A regular pentagram has the following Coxeter diagrams:

• x5/2o
• ß5o (as holosnub pentagon)

## 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+5)/2). Irregular pentagrams further appear as the vertex figures of some snub polyhedra.