# Pentagram

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Pentagram
Rank2
TypeRegular
Notation
Bowers style acronymStar
Coxeter diagramx5/2o ()
Schläfli symbol{5/2}
Elements
Edges5
Vertices5
Vertex figureDyad, length (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Inradius${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{20}}}\approx 0.16246}$
Area${\displaystyle {\frac {\sqrt {25-10{\sqrt {5}}}}{4}}\approx 0.40615}$
Angle36°
Central density2
Number of external pieces10
Level of complexity2
Related polytopes
ArmyPeg, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
DualPentagram
ConjugatePentagon
Convex corePentagon
Abstract & topological properties
Flag count10
Euler characteristic0
OrientableYes
Properties
SymmetryH2, order 10
Flag orbits1
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:

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

## 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.