Elongated pentagonal pyramid

Elongated pentagonal pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epeppy
Coxeter diagram	oxx5ooo&#xt
Elements
Faces	5 triangles, 5 squares, 1 pentagon
Edges	5+5+5+5
Vertices	1+5+5
Vertex figures	1 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	$\frac{5+\sqrt5+6\sqrt{25+10\sqrt5}}{24} ≈ 2.02198$
Dihedral angles	3–3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$
	3–4: $\arccos\left(-\sqrt{\frac{10-2\sqrt5}{15}}\right) ≈ 127.37737°$
	4–4: 108°
	4–5: 90°
Central density	1
Related polytopes
Army	Epeppy
Regiment	Epeppy
Dual	Elongated pentagonal pyramid
Conjugate	Elongated pentagrammic pyramid
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

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:

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

External links

• Klitzing, Richard. "epeppy".

• Quickfur. "The Elongated Pentagonal Pyramid".

• Wikipedia Contributors. "Elongated pentagonal pyramid".
• McCooey, David. "Elongated Pentagonal Pyramid"