# Decagon

(Redirected from Dec)
Decagon
Rank2
TypeRegular
Notation
Bowers style acronymDec
Coxeter diagramx10o ()
Schläfli symbol{10}
Elements
Edges10
Vertices10
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Inradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 1.53884}$
Area${\displaystyle 5{\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 7.69421}$
Angle144°
Central density1
Number of external pieces10
Level of complexity1
Related polytopes
ArmyDec
DualDecagon
ConjugateDecagram
Abstract & topological properties
Flag count20
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10), order 20
Flag orbits1
ConvexYes
NatureTame

The decagon is a polygon with 10 sides. A regular decagon has equal sides and equal angles. It has the most sides of any polygon that occurs as a face of a non-prismatic uniform polytope, although dodecagons appear in some tilings and hexadecagons appear in scaliform polychora.

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

It can be constructed as the uniform truncation of the regular pentagon. This is how it appears in many uniform polyhedra and polychora.

The decagon appears as the face of several compact regular skew polyhedra, and is the largest planar polygon to appear in a spherical, euclidean, compact hyperbolic polyhedron in 3D space.

## Naming

The name decagon is derived from the Ancient Greek δέκα (10) and γωνία (angle), referring to the number of vertices.

Other names include:

• Dec, Bowers style acronym, short for "decagon".

The combining prefix in BSAs is da-, as in dadip.

## Vertex coordinates

Coordinates for a regular decagon of unit edge length, centered at the origin are all sign changes of:

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

## Representations

A regular decagon can be represented by the following Coxeter diagrams:

• x10o () (regular),
• x5x () (H2 symmetry, generally a dipentagon),
• to5ot&#zx (t=${\displaystyle {\sqrt {\dfrac {5+{\sqrt {5}}}{2}}}}$, generally a pentambus),
• xFV Tto&#zx (rectangular symmetry, t as above, T=ft),
• xFVFx&#xt (axial edge-first),
• otTTto&#xt (axial vertex-first).

## Variations

Two main variants of the decagon have pentagon symmetry: the dipentagon, with two alternating side lengths and equal angles, and the dual pentambus, with two alternating angles and equal edges. Other less regular variations with chiral pentagonal, rectangular, central inversion, mirror, or no symmetry also exist.