geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a prism is a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

s, 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

prismatoid In geometry, a prismatoid is a polyhedron whose vertex (geometry), vertices all lie in two parallel Plane (geometry), planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces ...

s. Like many basic geometric terms, the word ''prism'' () 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).

Oblique vs right

An oblique prism is a prism in which the joining edges and faces are ''not

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

'' to the base faces. Example: a

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equiva ...

is an oblique prism whose base is a

, or equivalently a polyhedron with six parallelogram faces. Right Prism

A ''right'' prism is a prism in which the joining edges and faces are ''

'' to the base faces. This applies

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

all the joining faces are '' rectangular''. The dual of a ''right'' -prism is a ''right'' -

bipyramid In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...

. A right prism (with rectangular sides) with regular -gon bases has Schläfli symbol It approaches a cylinder as approaches

infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...

Special cases

*A right rectangular prism (with a rectangular base) is also called a ''

cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...

'', or informally a ''rectangular box''. A right rectangular prism has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

*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

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s. Thus all the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...

. They form one of the two infinite series of

semiregular polyhedra In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

, the other series being formed by the

antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

s. A uniform -gonal prism has

Properties

Volume

The

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of a prism is the product of the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

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 is the base area and is the height. The volume of a prism whose base is an -sided

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

with side length is therefore:

V = \frac h s^2 \cot\frac.

Surface area

The surface

of a right prism is: :

2B + Ph,

where is the area of the base, the height, and the base

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

. The surface area of a right prism whose base is a regular -sided

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

with side length , and with height , is therefore: :

A = \frac s^2 \cot\frac + nsh.

Symmetry

The

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of a right -sided prism with regular base is of order , except in the case of a cube, which has the larger symmetry group of order 48, which has three versions of as

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

s. The rotation group is of order , except in the case of a cube, which has the larger symmetry group of order 24, which has three versions of as subgroups. The symmetry group contains inversion

iff In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...

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

Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the ori ...
s

Similar polytopes

Truncated 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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

, and its sides are not parallelograms.

Twisted prism

A twisted prism is a nonconvex polyhedron constructed from a uniform -prism with each side face bisected on the square diagonal, by twisting the top, usually (but not necessarily) by radians ( degrees). If the bisectors are slanted to the left, then twisting the top base in the right direction (looking at the top of the prism) by a small angle gives nonconvex polyhedron and twisting it in the left direction, a convex polyhedron (see twisted square prism on the image). If the bisectors are slanted to the right, then twisting the top base in the left direction gives nonconvex polyhedron, in the right direction, convex one (see twisted dodecagonal prism). 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 -gonal ''twisted prism'' is topologically identical to the -gonal uniform

, but has half the

: , order . It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles. Any twisted -gonal prism is an antiprism, so the twisted square prism and twisted dodecagonal prism shown on the image are both antiprisms.

Frustum

A frustum is a similar construction to a prism, with

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

lateral faces and differently sized top and bottom polygons. Pentagonal frustum

Star prism

A star prism is a nonconvex polyhedron constructed by two identical

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A ''uniform star prism'' will have

with rectangles and 2 faces. It is topologically identical to a -gonal prism.

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 -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 -gonal prism.

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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

of zero. The topological polyhedral net can be cut from two rows of a

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

(with

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

): a band of squares, each attached to a crossed rectangle. An -gonal toroidal prism has vertices, faces: squares and crossed rectangles, and edges. It is topologically self-dual.

Prismatic polytope

A ''prismatic

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

'' is a higher-dimensional generalization of a prism. An -dimensional prismatic polytope is constructed from two ()-dimensional polytopes, translated into the next dimension. The prismatic -polytope elements are doubled from the ()-polytope elements and then creating new elements from the next lower element. Take an -polytope with -face elements (). Its ()-polytope prism will have -face elements. (With , .) By dimension: *Take a

with vertices, edges. Its prism has vertices, edges, and faces. *Take a

with vertices, edges, and faces. Its prism has vertices, edges, faces, and cells. *Take a polychoron with vertices, edges, faces, and cells. Its prism has vertices, edges, faces, cells, and hypercells.

Uniform prismatic polytope

A regular -polytope represented by

can form a uniform prismatic ()-polytope represented by a

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of two Schläfli symbols: By dimension: *A 0-polytopic prism is a

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

, represented by an empty

*: *A 1-polytopic prism is a

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

, made from 2 translated line segments. It is represented as the product Schläfli symbol If it is

, symmetry can be reduced: *:Example: , Square, two parallel line segments, connected by two line segment ''sides''. *A

al prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon can construct a uniform -gonal prism represented by the product If , with square sides symmetry it becomes a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

: *:Example: , Pentagonal prism, two parallel

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

s 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 can construct the uniform polychoric prism, represented by the product If the polyhedron and the sides are cubes, it becomes a

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

: *:Example: , Dodecahedral prism, two parallel dodecahedra connected by 12 pentagonal prism ''sides''. *... 23%2C29-duoprism_stereographic_closeup

Higher order prismatic polytopes also exist as

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

s 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 with vertices, edges, square faces, -gon faces, -gon faces, and bounded by -gonal prisms and -gonal prisms. For example, a ''4-4 duoprism'' is a lower symmetry form of a

, as is a ''cubic prism''. (4-4 duoprism prism), (cube-4 duoprism) and (tesseractic prism) are lower symmetry forms of a 5-cube.

References

* Chapter 2: Archimedean polyhedra, prisma and antiprisms

External links

*
Paper models of prisms and antiprisms
Free nets of prisms and antiprisms
Paper models of prisms and antiprisms
Using nets generated by '' Stella'' {{Authority control Prismatoid polyhedra Uniform polyhedra