# Prism (geometry)

In geometry, a **prism** is a polyhedron comprising an *n*-sided polygonal 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 for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

Set of uniform prisms | |
---|---|

(A hexagonal prism is shown) | |

Type | uniform polyhedron |

Conway polyhedron notation | Pn |

Faces | 2+n total:2 {n} n {4} |

Edges | 3n |

Vertices | 2n |

Schläfli symbol | {n}×{}[1] or t{2, n} |

Coxeter diagram | |

Vertex configuration | 4.4.n |

Symmetry group | D_{nh}, [n,2], (*n22), order 4n |

Rotation group | D_{n}, [n,2]^{+}, (n22), order 2n |

Dual polyhedron | bipyramids |

Properties | convex, semi-regular vertex-transitive |

n-gonal prism net (n = 9 here) |

## General, right and uniform prisms

A **right prism** is a prism in which the joining edges and faces are perpendicular to the base faces.[2] This applies if the joining faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, it is called an **oblique prism**.

For example a parallelepiped is an *oblique prism* of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms.

A **truncated prism** is a prism with nonparallel top and bottom faces.[3]

Some texts may apply the term **rectangular prism** or **square prism** to both a right rectangular-sided prism and a right square-sided prism. A *right p-gonal prism* with rectangular sides has a Schläfli symbol { } × {p}.

A right rectangular prism is also called a *cuboid*, or informally a *rectangular box*. A right square prism is simply a *square box*, and may also be called a *square cuboid*. A *right rectangular prism* has Schläfli symbol { }×{ }×{ }.

An *n*-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as *n* approaches infinity.

The term **uniform prism** or *semiregular prism* can be used for a *right prism* with square sides, since such prisms are in the set of uniform polyhedra. A *uniform p-gonal prism* has a Schläfli symbol t{2,p}. Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms.

## Volume

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

The volume is therefore:

where *B* is the base area and *h* is the height. The volume of a prism whose base is a regular *n*-sided polygon with side length *s* is therefore:

## Surface area

The surface area of a right prism is:

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 height *h* is therefore:

## Schlegel diagrams

P3 |
P4 |
P5 |
P6 |
P7 |
P8 |

## Symmetry

The symmetry group of a right *n*-sided prism with regular base is D_{nh} 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_{nh} contains inversion iff *n* is even.

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

## 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, and 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*}×{}.

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 {}×{}. If it is square, symmetry can be reduced: {}×{} = {4}.
Example: Square, {}×{}, 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*}×{}. If*p*= 4, with square sides symmetry it becomes a cube: {4}×{} = {4, 3}.Example: Pentagonal prism, {5}×{}, 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*}×{}. If the polyhedron is a cube, and the sides are cubes, it becomes a tesseract: {4, 3}×{} = {4, 3, 3}.Example: Dodecahedral prism, {5, 3}×{}, two parallel dodecahedra connected by 12 pentagonal prism *sides*.

- ...

Higher order prismatic polytopes also exist as cartesian products of any two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of these exist in 4-dimensional space are called duoprisms as the product of two polygons. Regular duoprisms are represented as {*p*}×{*q*}.

Family of uniform prisms | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Polyhedron | |||||||||||

Coxeter | |||||||||||

Tiling | |||||||||||

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 |

## Twisted prism

A **twisted prism** is a nonconvex prism polyhedron constructed by a uniform *q*-prism with the side faces bisected on the square diagonal, and twisting the top, usually by π/*q* radians (180/*q* degrees) in the same direction, causing side triangles to be concave.[4][5]

A twisted prism cannot be dissected into tetrahedra without adding new vertices. The smallest case, triangular form, is called a Schönhardt polyhedron.

A *twisted prism* is topologically identical to the antiprism, but has half the symmetry: D_{n}, [*n*,2]^{+}, order 2*n*. It can be seen as a convex 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 topologically identical to a prism, with trapezoid lateral faces and different 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*} × { }, with *p* rectangle and 2 {*p*/*q*} faces. It is topologically identical to a *p*-gonal prism.

{ }×{ }_{180}×{ } |
t_{a}{3}×{ } |
{5/2}×{ } | {7/2}×{ } | {7/3}×{ } | {8/3}×{ } | |
---|---|---|---|---|---|---|

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 base vertices are inverted around the center (or rotated 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an *p*-gonal hour glass, with all vertical edges passing through a single center, but no vertex is there. It is topologically identical to a *p*-gonal prism.

{ }×{ }_{180}×{ }_{180} |
t_{a}{3}×{ }_{180} |
{3}×{ }_{180} |
{4}×{ }_{180} |
{5}×{ }_{180} |
{5/2}×{ }_{180} |
{6}×{ }_{180} | |
---|---|---|---|---|---|---|---|

D_{2h}, order 8 |
D_{3d}, order 12 |
D_{4h}, order 16 |
D_{5d}, order 20 |
D_{6d}, order 24 | |||

### Toroidal prisms

A **toroidal prism** is a nonconvex polyhedron is like a *crossed prism* except instead of having base and top polygons, simple rectangular side faces are added to close 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 figure *4.4.4.4*. A *n*-gonal toroidal prism has 2*n* vertices and faces, and 4*n* edges and is topologically self-dual.

D_{4h}, order 16 |
D_{6h}, order 24 |

v=8, e=16, f=8 | v=12, e=24, f=12 |

## See also

- Apeirogonal prism
- Rectified prism
- Prismanes
- List of shapes

## References

- N.W. Johnson:
*Geometries and Transformations*, (2018) ISBN 978-1-107-10340-5 Chapter 11:*Finite symmetry groups*, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b - William F. Kern, James R Bland,
*Solid Mensuration with proofs*, 1938, p.28 - William F. Kern, James R Bland,
*Solid Mensuration with proofs*, 1938, p.81 - The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172

- 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

## External links

Wikisource has the text of the 1911 Encyclopædia Britannica article .Prism |

- Weisstein, Eric W. "Prism".
*MathWorld*. - Paper models of prisms and antiprisms Free nets of prisms and antiprisms
- Paper models of prisms and antiprisms Using nets generated by
*Stella*