# Pentagonal antiprism

Pentagonal antiprism
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymPap
Info
Coxeter diagrams2s10o
SymmetryI2(10)×A1+, order 20
ArmyPap
RegimentPap
Elements
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, (1+5)/2
Faces10 triangles, 2 pentagons
Edges10+10
Vertices10
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106}$
Volume${\displaystyle \frac{5+2\sqrt5}{6} ≈ 1.57869}$
Dihedral angles3–3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°}$
5–3: ${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°}$
Height${\displaystyle \sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065}$
Central density1
Euler characteristic2
Number of pieces12
Level of complexity4
Related polytopes
DualPentagonal antitegum
ConjugatePentagrammic retroprism
Properties
ConvexYes
OrientableYes
NatureTame

The pentagonal antiprism, or pap, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagons. Each vertex joins one pentagon and three triangles. As the name suggests, it is an antiprism based on a pentagon.

It can also be obtained as a diminishing of the regular icosahedron when two pentagonal pyramids are removed from opposite ends.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,+\frac12,\,0\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle ±\left(\frac12,\,0,\,\frac{1+\sqrt5}{2}\right).}$

These coordinates are obtained by removing two opposite vertices from a regular icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points:

• ${\displaystyle ±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),}$
• ${\displaystyle ±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),}$
• ${\displaystyle ±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{40}}\right).}$

## Representations

A pentagonal antiprism has the following Coxeter diagrams:

## General variant

The pentagonal antiprism has a general isogonal variant of the form xo5ox&#y that maintains its full symmetry. This veriant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by ${\displaystyle \sqrt{l^2-b^2\frac{5-\sqrt5}{10}}}$.

The bases of the pentagonal antiprism are rotated from each other by an angle of 36°. If this angle is changed the result is more properly called a pentagonal gyroprism.

A notable case occurs as the alternation of the uniform decagonal prism. This specific case has base edges of length ${\displaystyle \sqrt{\frac{5+\sqrt5}{2}}}$ and side edges of length ${\displaystyle \sqrt2}$.

## Related polyhedra

A pentagonal pyramid can be attached to a base of the pentagonal antiprism to form the gyroelongated pentagonal pyramid. If a second pyramid is attached to the other base, the result is the gyroelongated pentagonal bipyramid, better known as the regular icosahedron.

Two non-prismatic uniform polyhedron compounds are composed of pentagonal antiprisms:

There are also an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of pentagons.