# Pentagrammic pyramid

Pentagrammic pyramid
Rank3
TypeSegmentotope
SpaceSpherical
Notation
Bowers style acronymStappy
Coxeter diagramox5/2oo&#x
Elements
Faces5 triangles, 1 pentagram
Edges5+5
Vertices1+5
Vertex figures1 pentagram, edge length 1
5 isosceles triangles, edge lengths 1, 1, (5-1)/2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5-\sqrt{5}}{8}} ≈ 0.58779}$
Volume${\displaystyle \frac{5-\sqrt{5}}{24} ≈ 0.11516}$
Dihedral angles3-5/2: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt{5}}{15}}\right) ≈ 79.18768°}$
3-3: ${\displaystyle \arccos\left(\frac{\sqrt{5}}{3}\right) ≈ 41.81031°}$
Height${\displaystyle \sqrt{\frac{5+\sqrt{5}}{10}} ≈ 0.85065}$
Related polytopes
ArmyNon-RF peppy
RegimentStappy
DualPentagrammic pyramid
ConjugatePentagonal pyramid
Convex corePentagonal pyramid
Topological properties
OrientableYes
Properties
SymmetryH2×I, order 10
ConvexNo
NatureTame

The pentagrammic pyramid, or stappy, is a pyramid with a pentagrammic base and 5 triangles as sides.

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

## Vertex coordinates

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

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

## Related polyhedra

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