# Octagon

Octagon
Rank2
TypeRegular
Notation
Bowers style acronymOc
Coxeter diagramx8o ()
Schläfli symbol{8}
Elements
Edges8
Vertices8
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {2+{\sqrt {2}}}{2}}}\approx 1.30656}$
Inradius${\displaystyle {\frac {1+{\sqrt {2}}}{2}}\approx 1.20711}$
Area${\displaystyle 2(1+{\sqrt {2}})\approx 4.82843}$
Angle135°
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyOc
DualOctagon
ConjugateOctagram
Abstract & topological properties
Flag count16
Euler characteristic0
OrientableYes
Properties
SymmetryI2(8), order 16
Flag orbits1
ConvexYes
NatureTame

The octagon is a polygon with 8 sides. A regular octagon has equal sides and equal angles.

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

It can also be constructed as a uniform truncation of the square. It appears in higher uniform polytopes with hypercube symmetry in this form.

## Naming

The name octagon is derived from the Ancient Greek ὀκτώ (8) and γωνία (angle), referring to the number of vertices.

Other names include:

• Oc, Bowers style acronym, short for "octagon".
• 8-gon

The combining prefix in BSAs is o-, as in odip.

## Vertex coordinates

Coordinates for a regular octagon of unit edge length, centered at the origin, are all permutations of

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

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

• x8o () (full symmetry)
• x4x () (B2 symmetry, generally a ditetragon)
• ko4ok&#zx (B2, generally a tetrambus)
• xw wx&#zx (digonal symmetry)
• okK Kko#&zx (digonal symmetry, K=qk)
• xwwx&#xt (axial edge-first)
• okKko&#xt (axial vertex-first)

## Variations

Two main variants of the octagon have square symmetry: the ditetragon, with two alternating side lengths and equal angles, and the dual tetrambus, with two alternating angles and equal edges. Other less regular variations with chiral square, rectangular, inversion, mirror, or no symmetry also exist.

### Skew octagons

There are 12 regular skew octagons in Euclidean space.

Octagons in Euclidean space
Name Extended Schläfli symbol Dimensions
octagon ${\displaystyle \left\{{\dfrac {8}{1}}\right\}}$ 2
octagram ${\displaystyle \left\{{\dfrac {8}{3}}\right\}}$ 2
octagonal-square coil ${\displaystyle \left\{{\dfrac {8}{1,2}}\right\}}$ 4
octagonal-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,3}}\right\}}$ 4
skew octagon ${\displaystyle \left\{{\dfrac {8}{1,4}}\right\}}$ 3
square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{2,3}}\right\}}$ 4
skew octagram ${\displaystyle \left\{{\dfrac {8}{3,4}}\right\}}$ 3
octagonal-square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,2,3}}\right\}}$ 6
skew octagonal-square coil ${\displaystyle \left\{{\dfrac {8}{1,2,4}}\right\}}$ 5
skew octagonal-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,3,4}}\right\}}$ 5
skew square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{2,3,4}}\right\}}$ 5
skew octagonal-square-octagrammic coil ${\displaystyle \left\{{\dfrac {8}{1,2,3,4}}\right\}}$ 7