Prism (geometry)

Set of uniform $n$ -gonal prisms
Set of uniform n-gonal prisms
	Example: uniform hexagonal prism (n = 6)
Type	uniform in the sense of semiregular polyhedron
Faces	2 n-sided regular polygons; n squares
Edges	3n
Vertices	2n
Vertex configuration	4.4.n
Schläfli symbol	{n}×{ } ; t{2,n}
Conway notation	Pn
Coxeter diagram
Symmetry group	Dnh, [n,2], (*n22), order 4n
Rotation group	Dn, [n,2]+, (n22), order 2n
Dual polyhedron	convex dual-uniform n-gonal bipyramid
Properties	convex, regular polygon faces, isogonal, translated bases, sides ⊥ bases
Net
	Example: net of uniform enneagonal prism (n = 9)

In geometry, a prism is a polyhedron comprising an $n$ -sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and $n$ other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.^[2]

Like many basic geometric terms, the word prism (from Greek πρίσμα (prisma) 'something sawed') was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers).^[3]^[4]

Oblique vs right

An oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces.

Example: a parallelepiped is an oblique prism whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces.

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.^[5] This applies if and only if all the joining faces are rectangular.

The dual of a right $n$ -prism is a right $n$ -bipyramid.

A right prism (with rectangular sides) with regular $n$ -gon bases has Schläfli symbol ${ }\times{n}.$ It approaches a cylinder as $n$ approaches infinity.^[6]

Special cases

A right rectangular prism (with a rectangular base) is also called a cuboid, or informally a rectangular box. A right rectangular prism has Schläfli symbol ${ }\times{ }\times{ }.$
A right square prism (with a square base) is also called a square cuboid, or informally a square box.

Note: some texts may apply the term rectangular prism or square prism to both a right rectangular-based prism and a right square-based prism.

Types

Regular prism

A regular prism is a prism with regular bases.

Uniform prism

A uniform prism or semiregular prism is a right prism with regular bases and all edges of the same length.

Thus all the side faces of a uniform prism are squares.

Thus all the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are uniform polyhedra. They form one of the two infinite series of semiregular polyhedra, the other series being formed by the antiprisms.

A uniform $n$ -gonal prism has Schläfli symbol $t{2, n}.$

Family of uniform n-gonal prisms v t e
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.	2.4.4	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	...	∞.4.4
Coxeter diagram												...

Properties

Volume

The volume of a prism is the product of the area of the base by the height, i.e. the distance between the two base faces (in the case of a non-right prism, note that this means the perpendicular distance).

The volume is therefore:

V=Bh,

where $B$ is the base area and $h$ is the height.

The volume of a prism whose base is an $n$ -sided regular polygon with side length $s$ is therefore:

V={\frac {n}{4}}hs^{2}\cot {\frac {\pi }{n}}.

Surface area

The surface area of a right prism is:

2B+Ph,

where $B$ is the area of the base, $h$ the height, and $P$ the base perimeter.

The surface area of a right prism whose base is a regular $n$ -sided polygon with side length $s$ , and with height $h$ , is therefore:

A={\frac {n}{2}}s^{2}\cot {\frac {\pi }{n}}+nsh.

Symmetry

The symmetry group of a right $n$ -sided prism with regular base is $D n h$ of order $4 n$ , except in the case of a cube, which has the larger symmetry group $O h$ of order 48, which has three versions of $D 4h$ as subgroups. The rotation group is $D n$ of order $2 n$ , except in the case of a cube, which has the larger symmetry group $O$ of order 24, which has three versions of $D 4$ as subgroups.

The symmetry group $D n h$ contains inversion iff $n$ is even.

The hosohedra and dihedra also possess dihedral symmetry, and an $n$ -gonal prism can be constructed via the geometrical truncation of an $n$ -gonal hosohedron, as well as through the cantellation or expansion of an $n$ -gonal dihedron.

Schlegel diagrams

P3	P4	P5	P6	P7	P8

Similar polytopes

Truncated prism

Example truncated triangular prism. Its top face is truncated at an oblique angle, but it is not an oblique prism.

A truncated prism is formed when prism is sliced by a plane that is not parallel to its bases. A truncated prism's bases are not congruent, and its sides are not parallelograms.^[7]

Twisted prism

A twisted prism is a nonconvex polyhedron constructed from a uniform $n$ -prism with each side face bisected on the square diagonal, by twisting the top, usually by $π$ /n radians (180/n degrees) in the same direction, causing sides to be concave.^[8]^[9]

A twisted prism cannot be dissected into tetrahedra without adding new vertices. The simplest twisted prism has triangle bases and is called a Schönhardt polyhedron.

An $n$ -gonal twisted prism is topologically identical to the $n$ -gonal uniform antiprism, but has half the symmetry group: $D n, [n,2] +$ , order $2 n$ . It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

3-gonal	4-gonal		12-gonal
Schönhardt polyhedron	Twisted square prism	Square antiprism	Twisted dodecagonal antiprism

Frustum

A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.

Star prism

A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A uniform star prism will have Schläfli symbol ${p / q} \times { },$ with $p$ rectangles and 2 ${p / q}$ faces. It is topologically identical to a $p$ -gonal prism.

Examples
${ }\times{ } 180 \times{ }$	$t a {3}\times{ }$		${5/2}\times{ }$	${7/2}\times{ }$	${7/3}\times{ }$	${8/3}\times{ }$
$D 2h$ , order 8	$D 3h$ , order 12		$D 5h$ , order 20	$D 7h$ , order 28		$D 8h$ , order 32

Crossed prism

A crossed prism is a nonconvex polyhedron constructed from a prism, where the vertices of one base are inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an $n$ -gonal hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an $n$ -gonal prism.

Examples
${ }\times{ } 180 \times{ } 180$	$t a {3}\times{ } 180$		${3}\times{ } 180$	${4}\times{ } 180$	${5}\times{ } 180$	${5/2}\times{ } 180$	${6}\times{ } 180$
$D 2h$ , order 8	$D 3d$ , order 12			$D 4h$ , order 16	$D 5d$ , order 20		$D 6d$ , order 24

Toroidal prism

A toroidal prism is a nonconvex polyhedron like a crossed prism, but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for even-sided base polygons. These are topological tori, with Euler characteristic of zero. The topological polyhedral net can be cut from two rows of a square tiling (with vertex configuration $4.4.4.4$ ): a band of $n$ squares, each attached to a crossed rectangle. An $n$ -gonal toroidal prism has $2 n$ vertices, $2 n$ faces: $n$ squares and $n$ crossed rectangles, and $4 n$ edges. It is topologically self-dual.

Examples
$D 4h$ , order 16	$D 6h$ , order 24
$V = 8$ , $E = 16$ , $F = 8$	$V = 12$ , $E = 24$ , $F = 12$

Prismatic polytope

A prismatic polytope is a higher-dimensional generalization of a prism. An $n$ -dimensional prismatic polytope is constructed from two ( $n - 1$ )-dimensional polytopes, translated into the next dimension.

The prismatic $n$ -polytope elements are doubled from the ( $n - 1$ )-polytope elements and then creating new elements from the next lower element.

Take an $n$ -polytope with $F i$ $i$ -face elements ( $i = 0, ..., n$ ). Its ( $n + 1$ )-polytope prism will have $2 F i + F i -1$ $i$ -face elements. (With $F -1 = 0$ , $F n = 1$ .)

By dimension:

Take a polygon with $n$ vertices, $n$ edges. Its prism has $2 n$ vertices, $3 n$ edges, and $2 + n$ faces.
Take a polyhedron with $V$ vertices, $E$ edges, and $F$ faces. Its prism has $2 V$ vertices, $2 E + V$ edges, $2 F + E$ faces, and $2 + F$ cells.
Take a polychoron with $V$ vertices, $E$ edges, $F$ faces, and $C$ cells. Its prism has $2 V$ vertices, $2 E + V$ edges, $2 F + E$ faces, $2 C + F$ cells, and $2 + C$ hypercells.

Uniform prismatic polytope

A regular $n$ -polytope represented by Schläfli symbol ${p, q,..., t}$ can form a uniform prismatic ( $n + 1$ )-polytope represented by a Cartesian product of two Schläfli symbols: ${p, q,..., t}\times{ }.$

By dimension:

A 0-polytopic prism is a line segment, represented by an empty Schläfli symbol ${ }.$
A 1-polytopic prism is a rectangle, made from 2 translated line segments. It is represented as the product Schläfli symbol ${ }\times{ }.$ If it is square, symmetry can be reduced: ${ }\times{ } = {4}.$
Example: , Square, ${ }\times{ },$ two parallel line segments, connected by two line segment sides.
A polygonal prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon ${p}$ can construct a uniform $n$ -gonal prism represented by the product ${p}\times{ }.$ If $p = 4$ , with square sides symmetry it becomes a cube: ${4}\times{ } = {4,3}.$
Example: , Pentagonal prism, ${5}\times{ },$ two parallel pentagons connected by 5 rectangular sides.
A polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron ${p, q}$ can construct the uniform polychoric prism, represented by the product ${p, q}\times{ }.$ If the polyhedron and the sides are cubes, it becomes a tesseract: ${4,3}\times{ } = {4,3,3}.$
Example: , Dodecahedral prism, ${5,3}\times{ },$ two parallel dodecahedra connected by 12 pentagonal prism sides.
...

Higher order prismatic polytopes also exist as cartesian products of any two or more polytopes. The dimension of a product polytope is the sum of the dimensions of its elements. The first examples of these exist in 4-dimensional space; they are called duoprisms as the product of two polygons in 4-dimensions.

Regular duoprisms are represented as ${p}\times{q},$ with $pq$ vertices, $2 pq$ edges, $pq$ square faces, $p$ $q$ -gon faces, $q$ $p$ -gon faces, and bounded by $p$ $q$ -gonal prisms and $q$ $p$ -gonal prisms.

For example, ${4}\times{4},$ a 4-4 duoprism is a lower symmetry form of a tesseract, as is ${4,3}\times{ },$ a cubic prism. ${4}\times{4}\times{ }$ (4-4 duoprism prism), ${4,3}\times{4}$ (cube-4 duoprism) and ${4,3,3}\times{ }$ (tesseractic prism) are lower symmetry forms of a 5-cube.

References

^ Johnson, N. W (2018). "Chapter 11: Finite symmetry groups". Geometries and Transformations. ISBN 978-1-107-10340-5. See 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b.
^ Grünbaum, Branko (1997). "Isogonal Prismatoids". Discrete & Computational Geometry. 18: 13–52. doi:10.1007/PL00009307.
^ Malton, Thomas (1774). A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics. author, and sold. p. 360.
^ Elliot, James (1845). Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules. Longman, Brown, Green, and Longmans. p. 3.
^ Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. p. 28.
^ Geretschlager, Robert (2020). Engaging Young Students In Mathematics Through Competitions: World Perspectives And Practices. Vol. 1. World Scientific. p. 39. ISBN 978-981-120-582-8.
^ Kern & Bland (1938), p. 81.
^ Gorini, Catherine A. (2003). The facts on file: Geometry handbook. p. 172. ISBN 0-8160-4875-4.
^ "Pictures of Twisted Prisms".

Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisma and antiprisms