# Elongated pentagonal pyramid

Elongated pentagonal pyramid
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEpeppy
Coxeter diagramoxx5ooo&#xt
Elements
Faces5 triangles, 5 squares, 1 pentagon
Edges5+5+5+5
Vertices1+5+5
Vertex figures1 pentagon, edge length 1
5 kites, edge lengths 1 and 2
5 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle \frac{5+\sqrt5+6\sqrt{25+10\sqrt5}}{24} ≈ 2.02198}$
Dihedral angles3–3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°}$
3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{10-2\sqrt5}{15}}\right) ≈ 127.37737°}$
4–4: 108°
4–5: 90°
Central density1
Related polytopes
ArmyEpeppy
RegimentEpeppy
DualElongated pentagonal pyramid
ConjugateElongated pentagrammic pyramid
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal pyramid is one of the 92 Johnson solids (J9). It consists of 5 triangles, 5 squares, and 1 pentagon. It can be constructed by attaching a pentagonal prism to the base of the pentagonal pyramid..

If a second pyramid is attached to the other base of the pentagonal prism, the result is the elongated pentagonal bipyramid.

## Vertex coordinates

An elongated pentagonal pyramid of edge length 1 has the following vertices:

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