Pentagonal pyramid

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.

Pentagonal pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Peppy
Coxeter diagram	ox5oo&#x
Elements
Faces	5 triangles, 1 pentagon
Edges	5+5
Vertices	1+5
Vertex figures	1 pentagon, edge length 1
	5 isosceles triangles, edge lengths 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106$
Volume	$\frac{5+\sqrt5}{24} ≈ 0.30150$
Dihedral angles	3-3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$
	3-5: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°$
Height	$\sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573$
Central density	1
Related polytopes
Army	Peppy
Regiment	Peppy
Dual	Pentagonal pyramid
Conjugate	Pentagrammic pyramid
Abstract & topological properties
Flag count	40
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Net count	15
Nature	Tame

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:

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

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:

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

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 $\sqrt{l^2-b^2\frac{5+\sqrt5}{10}}$, its circumradius by $\frac{l}{2\sqrt{1-\frac{(5+\sqrt5)b^2}{10l^2}}}$, and its volume is given by $\frac{\sqrt{25+10\sqrt5}}{12}b^2\sqrt{l^2-b^2\frac{5+\sqrt5}{10}}$.

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.

External links

• Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#4 under ike).

• Klitzing, Richard. "peppy".

• Quickfur. "The Pentagonal Pyramid".

• Weisstein, Eric W. "Pentagonal Pyramid" ("Johnson solid") at MathWorld.

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

• Hi.gher.Space Wiki Contributors. "Pentagonal pyramid".