# Decagram

Decagram
Rank2
TypeRegular
Notation
Bowers style acronymDag
Coxeter diagramx10/3o ()
Schläfli symbol{10/3}
Elements
Edges10
Vertices10
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Inradius${\displaystyle {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 0.36327}$
Area${\displaystyle 5{\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\approx 1.81636}$
Angle72°
Central density3
Number of external pieces20
Level of complexity2
Related polytopes
ArmyDec, edge length ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$
DualDecagram
ConjugateDecagon
Convex coreDecagon
Abstract & topological properties
Flag count20
Euler characteristic0
Schläfli type{10}
OrientableYes
Properties
SymmetryI2(10), order 20
ConvexNo
NatureTame

The decagram is a star polygon with 10 sides. A regular decagram has equal sides and equal angles.

It is the second stellation of the decagon, and the only one that is not a compound. The only other polygons with a single non-compound stellation are the pentagon, the octagon, and the dodecagon.

It is the uniform quasitruncation of the pentagram, and as such appears frequently in uniform polyhedra and polychora. It is the largest uniform star polygon to appear in a non-prismatic uniform polytope in 3 or 4 dimensions.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{2}},\,0\right)}$.

## Representations

A decagram has the following Coxeter diagrams:

• x10/3o () (full symmetry)
• x5/3x () (H2 symmetry)