# Octagonal antiprism

Octagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymOap
Coxeter diagrams2s16o
Conway notationA8
Elements
Faces16 triangles, 2 octagons
Edges16+16
Vertices16
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 2+2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {6+2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}\approx 1.37555}$
Volume${\displaystyle {\frac {2{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}}{3}}\approx 4.26796}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2+{\sqrt {2}}}}}{3}}\right)\approx 153.96238^{\circ }}$
8–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {7+4{\sqrt {2}}-2{\sqrt {20+14{\sqrt {2}}}}}{3}}}\right)\approx 96.59451^{\circ }}$
Height${\displaystyle {\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{2}}}\approx 0.86030}$
Central density1
Number of external pieces18
Level of complexity4
Related polytopes
ArmyOap
RegimentOap
DualOctagonal antitegum
ConjugatesOctagrammic antiprism, Octagrammic retroprism
Abstract & topological properties
Flag count128
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(16)×A1)/2, order 32
ConvexYes
NatureTame

The octagonal antiprism, or oap, is a prismatic uniform polyhedron. It consists of 16 triangles and 2 octagons. Each vertex joins one octagon and three triangles. As the name suggests, it is an antiprism based on an octagon.

## Vertex coordinates

An octagonal antiprism of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,H\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,H\right),}$
• ${\displaystyle \left(0,\,\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,-H\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {2+{\sqrt {2}}}{2}}},\,0,\,-H\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,\pm {\frac {\sqrt {2+{\sqrt {2}}}}{2}},\,-H\right),}$

where ${\displaystyle H={\sqrt {\frac {-2-2{\sqrt {2}}+{\sqrt {20+14{\sqrt {2}}}}}{8}}}}$ is the distance between the antiprism's center and the center of one of its bases.

## Representations

An octagonal prism can be represented by the following Coxeter diagrams:

## Related polyhedra

A square cupola can be attached to a base of the octagonal antiprism to form the gyroelongated square cupola. If a second square cupola is attached to the other base, the result is the gyroelongated square bicupola.