Elongated pentagonal cupola

Elongated pentagonal cupola
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Epcu
Coxeter diagram	oxx5xxx&#xt
Elements
Faces	5 triangles, 5+5+5 squares, 1 pentagon, 1 decagon
Edges	5+5+5+5+5+10+10
Vertices	5+10+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	10 trapezoids, edge lengths 1, √2, √2, √2
	10 isosceles triangles, edge lengths √(5+√5)/2, √2, √2
Measures (edge length 1)
Volume
Dihedral angles	3–4 cupolaic:
	4–5:
	4–4 prismatic: 144°
	3–4 join:
	4–4 join:
	4–10: 90°
Central density	1
Related polytopes
Army	Epcu
Regiment	Epcu
Dual	Decakis order-10 truncated semibisected pentagonal trapezohedron
Conjugate	Elongated retrograde pentagrammic cupola
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The elongated pentagonal cupola is one of the 92 Johnson solids (J₂₀). It consists of 5 triangles, 5+5+5 squares, 1 pentagon, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal cupola.

If a second cupola is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobicupola. If the second cupola is rotated 36º instead, the result is the elongated pentagonal gyrobicupola.

Vertex coordinates[edit | edit source]

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

$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$
$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$
$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),$
$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),$
$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),$
$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),$