# Heptagonal prism

Heptagonal prism
Rank3
TypeUniform
Notation
Bowers style acronymHep
Coxeter diagramx x7o ()
Conway notationP7
Elements
Faces7 squares, 2 heptagons
Edges7+14
Vertices14
Vertex figureIsosceles triangle, edge lengths 2, 2, 2cos(π/7)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {\pi }{7}}}}}}{2}}\approx 1.25618}$
Volume${\displaystyle {\frac {7}{4\tan {\frac {\pi }{7}}}}\approx 3.63391}$
Dihedral angles4–4: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
4–7: 90°
Height1
Central density1
Number of external pieces9
Level of complexity3
Related polytopes
ArmyHep
RegimentHep
DualHeptagonal tegum
ConjugatesHeptagrammic prism, Great heptagrammic prism
Abstract & topological properties
Flag count84
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonGP(7,1)
Properties
SymmetryI2(7)×A1, order 28
ConvexYes
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:

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