# Hexagonal prism

Last updated
Uniform hexagonal prism
Type Prismatic uniform polyhedron
Elements F = 8, E = 18, V = 12 (χ = 2)
Faces by sides6{4}+2{6}
Schläfli symbol t{2,6} or {6}×{}
Wythoff symbol 2 6 | 2
2 2 3 |
Coxeter diagrams

Symmetry D6h, [6,2], (*622), order 24
Rotation group D6, [6,2]+, (622), order 12
References U 76(d)
Dual Hexagonal dipyramid
Properties convex, zonohedron

Vertex figure
4.4.6

In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices. [1]

## Contents

Since it has 8 faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and tilarity of the various eight-sided figures, the term is rarely used without clarification.

Before sharpening, many pencils take the shape of a long hexagonal prism. [2]

## As a semiregular (or uniform) polyhedron

If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron , represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right hexagonal prism is D6h of order 24. The rotation group is D6 of order 12.

## Volume

As in most prisms, the volume is found by taking the area of the base, with a side length of ${\displaystyle a}$, and multiplying it by the height ${\displaystyle h}$, giving the formula: [3]

${\displaystyle V={\frac {3{\sqrt {3}}}{2}}a^{2}\times h}$ and it's surface area can be ${\displaystyle S=3a({\sqrt {3}}a+2h)}$.

## Symmetry

The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:

Name Symmetry D6h, [2,6], (*622) C6v, [6], (*66) D3h, [2,3], (*322) Regular-hexagonal prism Hexagonal frustum Ditrigonal prism Triambic prism Ditrigonal trapezoprism

## As part of spatial tesselations

It exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

 Hexagonal prismatic honeycomb [1] Triangular-hexagonal prismatic honeycomb Snub triangular-hexagonal prismatic honeycomb Rhombitriangular-hexagonal prismatic honeycomb

It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:

 truncated tetrahedral prism truncated octahedral prism Truncated cuboctahedral prism Truncated icosahedral prism Truncated icosidodecahedral prism runcitruncated 5-cell omnitruncated 5-cell runcitruncated 16-cell omnitruncated tesseract runcitruncated 24-cell omnitruncated 24-cell runcitruncated 600-cell omnitruncated 120-cell
Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622)[6,2]+, (622)[6,2+], (2*3)
{6,2} t{6,2} r{6,2} t{2,6} {2,6} rr{6,2} tr{6,2} sr{6,2} s{2,6}
Duals to uniforms
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p< 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p> 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb.Paraco.Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*32
[,3]

[12i,3]

[9i,3]

[6i,3]

[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6. 4.6.24i4.6.18i4.6.12i4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.V4.6.24iV4.6.18iV4.6.12iV4.6.6i

Family of uniform n-gonal prisms
Prism name Digonal prism (Trigonal)
Triangular prism
(Tetragonal)
Square prism
Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism Enneagonal prism Decagonal prism Hendecagonal prism Dodecagonal prism ... Apeirogonal prism
Polyhedron image ...
Spherical tiling image Plane tiling image
Vertex config. 2.4.43.4.44.4.45.4.46.4.47.4.48.4.49.4.410.4.411.4.412.4.4...∞.4.4
Coxeter diagram ...

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## References

1. Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN   9780520030565 .
2. Simpson, Audrey (2011), Core Mathematics for Cambridge IGCSE, Cambridge University Press, pp. 266–267, ISBN   9780521727921 .
3. Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN   9781564149367 .