Hexagonal prism

Short description: Prism with a 6-sided base

Template:Prism even polyhedron stat table File:Prisma hexagonal 3D.stl

In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.^[1]

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 D_6h 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 [math]\displaystyle{ a }[/math], and multiplying it by the height [math]\displaystyle{ h }[/math], giving the formula:^[3]

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

Symmetry

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

Symmetry	D_6h, [2,6], (*622)	C_6v, [6], (*66)	D_3h, [2,3], (*322)		D_3d, [2⁺,6], (2*3)
Name	Regular-hexagonal prism	Hexagonal frustum	Ditrigonal prism	Triambic prism	Ditrigonal trapezoprism
Construction	{6}×{},		t{3}×{},		s₂{2,6},
Image
Distortion

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

Related polyhedra and tilings

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.