# Hexagonal pyramid

Hexagonal pyramid
Rank3
TypeCRF
SpaceEuclidean
Notation
Bowers style acronymHippy
Coxeter diagramox6oo&#x
Elements
Faces6 triangles, 1 hexagon
Edges6+6
Vertices1+6
Vertex figures1 hexagon, edge length 1
6 isosceles triangles, edge lengths 1, 1, 3
Measures (edge length 1)
Volume0
Dihedral angles3-3: 180º
3-6: 0º
Height0
Number of external pieces7
Level of complexity3
Related polytopes
ArmyHippy
RegimentHippy
DualHexagonal pyramid
ConjugateNone
Abstract & topological properties
Flag count48
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryG2×I, order 12
ConvexYes
Net count32
NatureTame

The hexagonal pyramid, or hippy, is a pyramid with a hexagonal base and 6 triangles as sides. The version with equilateral triangles as sides is flat, as a regular hexagon can be exactly decomposed into 6 equilateral triangles by a central point. Other variants with isosceles triangles as sides exist as non-degenerate polyhedra.

The flat variant with equilateral triangles is the vertex-first cap of the triangular tiling.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0\right),}$
• ${\displaystyle \left(\pm 1,\,0,\,0\right),}$
• ${\displaystyle \left(0,\,0,\,0\right).}$

These coordinates are a subset of the vertices of the regular triangular tiling.

## Representations

A hexagonal pyramid has the following Coxeter diagrams:

## General variant

For the general hexagonal pyramid with base edges of length b and lacing edges of length l, its height is given by ${\displaystyle {\sqrt {l^{2}-b^{2}}}}$, its circumradius by ${\displaystyle {\frac {l}{2{\sqrt {1-{\frac {b^{2}}{l^{2}}}}}}}}$, and its volume is given by ${\displaystyle {\frac {\sqrt {3}}{2}}b^{2}{\sqrt {l^{2}-b^{2}}}}$.