{{Short description|Prism with a 6-sided base}} {{infobox polyhedron | image = Hexagonal Prism.svg | name = Hexagon prism | symmetry = prismatic symmetry <math> D_{6\mathrm{h}} </math> of order 24 | type = prism,<br>parallelohedron | dual = hexagonal bipyramid }} thumb|3D model of a uniform hexagonal prism

In geometry, the '''hexagonal prism''' is a prism with hexagonal base. this polyhedron has 8 faces, 18 edges, and 12 vertices.

== Properties == A hexagonal prism has twelve vertices, eighteen edges, and eight faces. Every prism has two faces known as its bases, and the bases of a hexagonal prism are hexagons. The hexagons has six vertices, each of which pairs with another hexagon's vertex, forming six edges. These edges form three parallelograms as other faces.<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> A prism is said to be right if the edges are of the same length and perpendicular to the base.

If faces are all regular, the hexagonal prism is a semiregular polyhedron&mdash;more generally, a uniform polyhedron&mdash;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 symmetry group of a right hexagonal prism is prismatic symmetry <math> D_{6 \mathrm{h}} </math> of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.<ref>{{citation | last1 = Flusser | first1 = J. | last2 = Suk | first2 = T. | last3 = Zitofa | first3 = B. | year = 2017 | title = 2D and 3D Image Analysis by Moments | publisher = John Wiley & Sons | isbn = 978-1-119-03935-8 | url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 | page = 126 }}</ref> The dual of a hexagonal prism is a hexagonal bipyramid, both of which have the same three-dimensional symmetry group.

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 display="block"> V = \frac{3 \sqrt{3}}{2}a^2h, </math> and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares: <math display="block"> S = 3a(\sqrt{3}a+2h).</math>

== Honeycombs == thumb|upright=0.8|Hexagonal prismatic honeycomb The hexagonal prism is one of the parallelohedra, a polyhedral class that can be translated without rotations in Euclidean space, producing honeycombs; this class was discovered by Evgraf Fedorov in accordance with his studies of crystallography systems. The hexagonal prism is generated from four line segments, three of them parallel to a common plane and the fourth not.<ref name=alexandrov>{{citation|last=Alexandrov|first=A. D.|author-link=Aleksandr Danilovich Aleksandrov|contribution=8.1 Parallelohedra|pages=349–359|publisher=Springer|title=Convex Polyhedra|title-link=Convex Polyhedra (book)|year=2005}}</ref> Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.<ref name=onset>{{citation | last1 = Delaney | first1 = Gary W. | last2 = Khoury | first2 = David | date = February 2013 | doi = 10.1140/epjb/e2012-30445-y | issue = 2 | journal = The European Physical Journal B | title = Onset of rigidity in 3D stretched string networks | volume = 86| page = 44 | bibcode = 2013EPJB...86...44D }}</ref>

The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions: {| class=wikitable width=400 |- align=center |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 |100px |100px |100px |}

It also exists as cells of a number of four-dimensional uniform 4-polytopes, 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 |100px |100px |100px |100px |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}} |- |100px |100px |100px |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 |100px |100px |100px |100px |}

==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