# Pentagonal pyramid

Pentagonal pyramid
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymPeppy
Coxeter diagramox5oo&#x
Elements
Faces5 triangles, 1 pentagon
Edges5+5
Vertices1+5
Vertex figures1 pentagon, edge length 1
5 isosceles triangles, edge lengths 1, 1, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106}$
Volume${\displaystyle \frac{5+\sqrt5}{24} ≈ 0.30150}$
Dihedral angles3-3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°}$
3-5: ${\displaystyle \arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°}$
Height${\displaystyle \sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573}$
Central density1
Related polytopes
ArmyPeppy
RegimentPeppy
DualPentagonal pyramid
ConjugatePentagrammic pyramid
Abstract properties
Flag count40
Net count15
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The pentagonal pyramid, or peppy, is a pyramid with a pentagonal base and 5 triangles as sides. The version with equilateral triangles as sides is the second of the 92 Johnson solids (J2). In what follows, unless otherwise specified, this what will be meant by a "pentagonal pyramid", even though other variants with isosceles triangles as sides exist.

It is the vertex-first cap of the icosahedron. A regular icosahedron can be constructed by attaching two pentagonal pyramids to the bases of a pentagonal antiprism.

It is one of three regular polygonal pyramids to be CRF. The others are the regular tetrahedron (triangular pyramid) and the square pyramid.

## Vertex coordinates

A pentagonal pyramid of edge length 1 has the following vertices:

• ${\displaystyle \left(0,\,±\frac12,\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,\frac{1+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,0,\,\frac12\right).}$

These coordinates are obtained as a subset of the vertices of the regular icosahedron.

Alternatively, starting from the coordinates of a regular pentagon in the plane, we obtain the pyramid with the following coordinates:

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

## Related polyhedra

Two pentagonal pyramids can be attached at their bases to form a pentagonal tegum.

A pentagonal prism can be attached to the base of a pentagonal pyramid to form the elongated pentagonal pyramid. If a pentagonal antiprism is attached instead, the result is the gyroelongated pentagonal pyramid.

## General variant

For the general pentagonal pyramid with base edges of length b and lacing edges of length l, its height is given by ${\displaystyle \sqrt{l^2-b^2\frac{5+\sqrt5}{10}}}$, its circumradius by ${\displaystyle \frac{l}{2\sqrt{1-\frac{(5+\sqrt5)b^2}{10l^2}}}}$, and its volume is given by ${\displaystyle \frac{\sqrt{25+10\sqrt5}}{12}b^2\sqrt{l^2-b^2\frac{5+\sqrt5}{10}}}$.

Pentagonal pyramids occur as vertex figures of 6 uniform polychora, including the convex truncated hexacosichoron, truncated great hecatonicosachoron, truncated great faceted hexacosichoron, quasitruncated great stellated hecatonicosachoron, icosahedral prism, and small stellated dodecahedral prism.