# Heptagonal prism

Heptagonal prism Rank3
TypeUniform
SpaceSpherical
Bowers style acronymHep
Info
Coxeter diagramx x7o
SymmetryI2(7)×A1, order 28
ArmyHep
RegimentHep
Elements
Vertex figureIsosceles triangle, edge lengths 2, 2, 2cos(π/7)
Faces7 squares, 2 heptagons
Edges7+14
Vertices14
Measures (edge length 1)
Circumradius$\frac{\sqrt{1+\csc^2\frac\pi7}}{2} ≈ 1.25618$ Volume$\frac{7}{4\tan\frac\pi7} ≈ 3.63391$ Dihedral angles4–4: $\frac{5\pi}{7} ≈ 128.57143°$ 4–7: 90°
Height1
Central density1
Euler characteristic2
Number of pieces9
Level of complexity3
Related polytopes
DualHeptagonal tegum
ConjugatesHeptagrammic prism, Great heptagrammic prism
Properties
ConvexYes
OrientableYes
NatureTame

The heptagonal prism, or hep, is a prismatic uniform polyhedron. It consists of 2 heptagons and 7 squares. Each vertex joins one heptagon and two squares. As the name suggests, it is a prism based on a heptagon.

## Vertex coordinates

The coordinates of a heptagonal prism, centered at the origin and with edge length 2sin(π/7), are given by:

• $\left(1,\,0,\,±\sin\frac\pi7\right),$ • $\left(\cos\left(\frac{2\pi}{7}\right),\,±\sin\left(\frac{2\pi}{7}\right),\,±\sin\frac\pi7\right),$ • $\left(\cos\left(\frac{4\pi}{7}\right),\,±\sin\left(\frac{4\pi}{7}\right),\,±\sin\frac\pi7\right),$ • $\left(\cos\left(\frac{6\pi}{7}\right),\,±\sin\left(\frac{6\pi}{7}\right),\,±\sin\frac\pi7\right).$ 