View source for Hexagonal prism

{{Short description|Prism with a 6-sided base}}
{{Prism polyhedra db|Prism even polyhedron stat table|P6}}
[[File:Prisma hexagonal 3D.stl|thumb|3D model of a uniform hexagonal prism.]]

In [[geometry]], the '''hexagonal prism''' is a [[Prism (geometry)|prism]] with [[hexagon]]al base. Prisms are [[polyhedron]]s; this polyhedron has 8 [[face (geometry)|faces]], 18 [[Edge (geometry)|edges]], and 12 [[vertex (geometry)|vertices]].<ref name="pugh">{{citation|title=Polyhedra: A Visual Approach|first=Anthony|last=Pugh|publisher=University of California Press|year=1976|isbn=9780520030565|pages=21, 27, 62|url=https://books.google.com/books?id=IDDxpYQTR7kC&pg=PA21}}.</ref>

Since it has 8 faces, it is an [[octahedron#Related_polyhedra|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 [[pencil]]s take the shape of a long hexagonal prism.<ref>{{citation|title=Core Mathematics for Cambridge IGCSE|first=Audrey|last=Simpson|publisher=Cambridge University Press|year=2011|isbn=9780521727921|pages=266–267|url=https://books.google.com/books?id=Xm80FfQZ42AC&pg=PA266}}.</ref>

== 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 '''[[truncation (geometry)|truncated]] [[hosohedron|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 polyhedron|dual]] of a hexagonal prism is a [[hexagonal bipyramid]].

The [[symmetry group]] of a right hexagonal prism is [[dihedral group|''D<sub>6h</sub>'']] of order 24. The [[Point groups in three dimensions#Rotation groups|rotation group]] is ''D<sub>6</sub>'' of order 12.

== Volume==
As in most prisms, the volume is found by taking the area of the base, with a side length of <math>a</math>, and multiplying it by the height <math>h</math>, giving the formula:<ref>{{citation|title=Geometry|first=Carolyn C.|last=Wheater|publisher=Career Press|year=2007|isbn=9781564149367|pages=236–237}}</ref>

<math>V = \frac{3 \sqrt{3}}{2}a^2 \times h</math>
and its surface area can be
<math>S=3a(\sqrt{3}a+2h)</math>.

== Symmetry ==
The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:
{| class=wikitable width=480
|- align=center
!Name
|'''Regular-hexagonal prism'''
|'''Hexagonal frustum'''
|'''Ditrigonal prism'''
|'''Triambic prism'''
|'''Ditrigonal trapezoprism'''
|- align=center
![[List_of_spherical_symmetry_groups|Symmetry]]
!D<sub>6h</sub>, [2,6], (*622)
!C<sub>6v</sub>, [6], (*66)
!colspan=2|D<sub>3h</sub>, [2,3], (*322)
!D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)
|- align=center
!Construction
!{6}×{}, {{CDD|node_1|2|node_1|6|node}}
!
!t{3}×{}, {{CDD|node_1|2|node_1|3|node_1}}
!{{CDD|node_1|2|node_f1|3|node_f1}}
!s<sub>2</sub>{2,6}, {{CDD|node_h|2x|node_h|6|node_1}}
|-
!Image
|[[Image:Hexagonal Prism.svg|100px]]
|[[File:Hexagonal frustum.png|100px]]
|[[File:Truncated triangle prism.png|100px]]
|
|[[File:Cantic_snub_hexagonal_hosohedron.png|100px]]
|-
!Distortion
|
|[[File:Hexagonal frustum2.png|100px]]
|[[File:Truncated triangle prism2.png|100px]]
|[[File:Isohedral_hexagon_prism.png|60px]]<BR>[[File:Isohedral_hexagon_prism2.png|60px]]
|[[File:Cantic_snub_hexagonal_hosohedron2.png|100px]]
|}

== As part of spatial tesselations ==

It exists as cells of four prismatic [[uniform convex honeycomb]]s in 3 dimensions:
{| class=wikitable width=400
|- align=center
|[[Hexagonal prismatic honeycomb]]<ref name="pugh"/><BR>{{CDD|node_1|6|node|3|node|2|node_1|infin|node}}
|[[Triangular-hexagonal prismatic honeycomb]]<BR>{{CDD|node|6|node_1|3|node|2|node_1|infin|node}}
|[[Snub triangular-hexagonal prismatic honeycomb]]<BR>{{CDD|node_h|6|node_h|3|node_h|2|node_1|infin|node}}
|[[Rhombitriangular-hexagonal prismatic honeycomb]]<BR>{{CDD|node_1|6|node|3|node_1|2|node_1|infin|node}}
|- align=center
|[[File:Hexagonal_prismatic_honeycomb.png|100px]]
|[[File:Triangular-hexagonal_prismatic_honeycomb.png|100px]]
|[[File:Snub triangular-hexagonal prismatic honeycomb.png|100px]]
|[[File:Rhombitriangular-hexagonal prismatic honeycomb.png|100px]]
|}

It also exists as cells of a number of four-dimensional [[uniform 4-polytope]]s, including: 
{| class=wikitable width=500
|- align=center
|[[truncated tetrahedral prism]]<BR>{{CDD|node_1|3|node_1|3|node|2|node_1}}
|[[truncated octahedral prism]]<BR>{{CDD|node_1|3|node_1|4|node|2|node_1}}
|[[Truncated cuboctahedral prism]]<BR>{{CDD|node_1|3|node_1|4|node_1|2|node_1}}
|[[Truncated icosahedral prism]]<BR>{{CDD|node_1|3|node_1|5|node|2|node_1}}
|[[Truncated icosidodecahedral prism]]<BR>{{CDD|node_1|3|node_1|5|node_1|2|node_1}}
|- align=center
|[[File:Truncated tetrahedral prism.png|100px]]
|[[File:Truncated_octahedral_prism.png|100px]]
|[[File:Truncated cuboctahedral prism.png|100px]]
|[[File:Truncated icosahedral prism.png|100px]]
|[[File:Truncated icosidodecahedral prism.png|100px]]
|- align=center
|[[runcitruncated 5-cell]]<BR>{{CDD|node_1|3|node|3|node_1|3|node_1}}
|[[omnitruncated 5-cell]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1}}
|[[runcitruncated 16-cell]]<BR>{{CDD|node_1|4|node|3|node_1|3|node_1}}
|[[omnitruncated tesseract]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
|-
|[[File:4-simplex t013.svg|100px]]
|[[File:4-simplex t0123.svg|100px]]
|[[File:4-cube t023.svg|100px]]
|[[File:4-cube t0123.svg|100px]]
|- align=center
|[[runcitruncated 24-cell]]<BR>{{CDD|node_1|3|node|4|node_1|3|node_1}}
|[[omnitruncated 24-cell]]<BR>{{CDD|node_1|3|node_1|4|node_1|3|node_1}}
|[[runcitruncated 600-cell]]<BR>{{CDD|node_1|5|node|3|node_1|3|node_1}}
|[[omnitruncated 120-cell]]<BR>{{CDD|node_1|5|node_1|3|node_1|3|node_1}}
|- align=center
|[[File:24-cell t0123 F4.svg|100px]]
|[[File:24-cell t013 F4.svg|100px]]
|[[File:120-cell_t023_H3.png|100px]]
|[[File:120-cell_t0123_H3.png|100px]]
|}

== Related polyhedra and tilings ==
{{Hexagonal dihedral truncations}}

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and [[Coxeter-Dynkin diagram]] {{CDD|node_1|p|node_1|3|node_1}}.  For ''p'' &lt; 6, the members of the sequence are [[Omnitruncation (geometry)|omnitruncated]] polyhedra ([[zonohedron]]s), shown below as spherical tilings. For ''p'' &gt; 6, they are tilings of the hyperbolic plane, starting with the [[truncated triheptagonal tiling]].

{{Omnitruncated table}}

== See also ==
{{UniformPrisms}}

==References==
{{reflist}}

== External links ==
*[http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space] VRML models
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra [http://www.georgehart.com/virtual-polyhedra/prisms-index.html Prisms and antiprisms]
* {{mathworld | urlname = HexagonalPrism | title = Hexagonal prism}}
*[https://web.archive.org/web/20071008014242/http://polyhedra.org/poly/show/24/hexagonal_prism Hexagonal Prism Interactive Model] -- works in your web browser

[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]
{{Polyhedron-stub}}