# Decagonal antiprism

Decagonal antiprism
Rank3
TypeUniform
Notation
Bowers style acronymDap
Coxeter diagrams2s20o ()
Conway notationA10
Elements
Faces20 triangles, 2 decagons
Edges20+20
Vertices20
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {8+2{\sqrt {5}}+{\sqrt {50+22{\sqrt {5}}}}}{8}}}\approx 1.67450}$
Volume${\displaystyle {\frac {5{\sqrt {-2-2{\sqrt {5}}+2{\sqrt {650+290{\sqrt {5}}}}}}}{6}}\approx 6.74929}$
Dihedral angles3–3: ${\displaystyle \arccos \left({\frac {1-{\sqrt {10+2{\sqrt {5}}}}}{3}}\right)\approx 159.18651^{\circ }}$
10–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {11+4{\sqrt {5}}-2{\sqrt {50+22{\sqrt {5}}}}}{3}}}\right)\approx 95.24664^{\circ }}$
Height${\displaystyle {\sqrt {\frac {-4-2{\sqrt {5}}+{\sqrt {50+22{\sqrt {5}}}}}{2}}}\approx 0.86240}$
Central density1
Number of external pieces22
Level of complexity4
Related polytopes
ArmyDap
RegimentDap
DualDecagonal antitegum
ConjugateDecagrammic antiprism
Abstract & topological properties
Flag count160
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(20)×A1)/2, order 40
ConvexYes
NatureTame

The decagonal antiprism, or dap, is a prismatic uniform polyhedron. It consists of 20 triangles and 2 decagons. Each vertex joins one decagon and three triangles. As the name suggests, it is an antiprism based on a decagon.

## Vertex coordinates

A decagonal antiprism of edge length 1 has vertex coordinates given by:

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

where ${\displaystyle H={\sqrt {\frac {{\sqrt {50+22{\sqrt {5}}}}-2{\sqrt {5}}-4}{8}}}}$ is the distance between the antiprism's center and the center of one of its bases.

## Representations

A decagonal antiprism has the following Coxeter diagrams:

## Related polyhedra

A pentagonal cupola can be attached to a base of the decagonal antiprism to form the gyroelongated pentagonal cupola. If a pentagonal rotunda is attached instead the result is the gyroelongated pentagonal rotunda.

If two pentagonal cupolas are attached to the bases, the result is the gyroelongated pentagonal bicupola. If two rotundas are attached, the result is the gyroelongated pentagonal birotunda. If one cupola and one rotunda are attached, the result is the gyroelongated pentagonal cupolarotunda.