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, edge length ${\displaystyle \frac{\sqrt5-1}{2}}$
DualPentagram
ConjugatePentagon
Convex corePentagon
Abstract properties
Flag count10
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryH2, order 10
ConvexNo
NatureTame

The pentagram 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 regular 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.