geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a prism is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

comprising an polygon base, a second base which is a

translated Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

copy (rigidly moved without rotation) of the first, and other

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

, necessarily all

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

s, joining

corresponding sides In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, ta ...

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 In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

base is called a pentagonal prism. Prisms are a subclass of

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

s. Like many basic geometric terms, the word ''prism'' () was first used in

Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...

. 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 relation to the nature of the bases, which caused confusion among later geometry writers.

Oblique prism

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

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

'' 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. In Euclidea ...

is an oblique prism whose base is a

, or equivalently a polyhedron with six parallelogram faces.

Right prism, uniform prism

Right prism

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

'' to the base faces.William F. Kern, James R. Bland, ''Solid Mensuration with proofs'', 1938, p. 28. This applies

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

all the joining faces are ''

rectangular In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a par ...

''. The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of a ''right'' ''n''-prism is a ''right'' ''n''-

bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not ...

. A right prism (with rectangular sides) with regular ''n''-gon bases has Schläfli symbol ×. It approaches a

cylindrical A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infini ...

solid as ''n'' approaches

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

; a

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

is considered a circular prism.

Special cases

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

cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...

'', 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 regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

××. *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.

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 Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

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 (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

alright. 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 direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s. A uniform ''n''-gonal prism has

Volume

The

volume Volume is a measure of occupied 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). The de ...

of a prism is the product of the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

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 = B h ,

where ''B'' is the base area and ''h'' is the height. The volume of a prism whose base is an ''n''-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 p ...

with side length ''s'' is therefore: :

V = \frac h s^2 \cot \left( \frac \right) .

Surface area

The surface

of a right prism is: :

2 B + P h ,

where ''B'' is the area of the base, ''h'' the height, and ''P'' the base

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

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

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

with side length ''s'', and with height ''h'', is therefore: :

A = \frac s^2 \cot \left( \frac \right) + n s h .

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 origina ...
s

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 ambient ...

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

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s. The

rotation group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

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 D₄ as subgroups. The symmetry group D_''n''h contains

inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...

iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...

''n'' is even. The

hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...

and

dihedra A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

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 In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...

expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

of an ''n''-gonal dihedron.

Truncated prism

A truncated prism is a prism with non-

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

top and bottom faces. TruncatedTriangularPrism

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 radians ( degrees) in the same direction, causing sides to be concave. A twisted prism cannot be

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

into tetrahedra without adding new vertices. The smallest case: the triangular form, is called a

Schönhardt polyhedron In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928. The same ...

. An ''n''-gonal ''twisted prism'' is topologically identical to the ''n''-gonal uniform

, but has half the

: D_''n'', 'n'',2sup>+, order 2''n''. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

Frustum

frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are p ...

is a similar construction to a prism, with

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

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, certain notable ones can arise through truncation operations ...

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 ''p'' rectangle and 2 faces. It is topologically identical to a ''p''-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 rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containin ...

s. For a regular polygon base, the appearance is an ''n''-gonal

hour glass An hourglass (or sandglass, sand timer, sand clock or egg timer) is a device used to measure the passage of time. It comprises two glass bulbs connected vertically by a narrow neck that allows a regulated flow of a substance (historically sand) ...

. 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.

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 ...

of zero. The topological

polyhedral net In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and sol ...

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. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the s ...

(with

vertex configuration In geometry, a vertex configurationCrystallography ...

''4.4.4.4''): a band of ''n'' squares, each attached to a

. 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 In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...

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 -d ...

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

with ''n'' vertices, ''n'' edges. Its prism has 2''n'' vertices, 3''n'' edges, and faces. *Take a

with ''v'' vertices, ''e'' edges, and ''f'' faces. Its prism has 2''v'' vertices, edges, faces, and cells. *Take a

polychoron In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...

with ''v'' vertices, ''e'' edges, ''f'' faces, and ''c'' cells. Its prism has 2''v'' vertices, edges, faces, cells, and hypercells.

Uniform prismatic polytope

A regular ''n''-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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

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

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

, represented by an empty

. *: *A 1-polytopic prism is a

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

, 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 ''n''-gonal prism represented by the product ×. If , with square sides symmetry it becomes a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

: *:Example: ,

Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

, ×, two parallel

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

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 is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

: × = *:Example: ,

Dodecahedral prism In geometry, a dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces: 30 squares and 24 pentagons. It has 80 edges and 40 vertices. It can be ...

, ×, two parallel

dodecahedra In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

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

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

s as the product of two polygons in 4-dimensions. Regular duoprisms are represented as ×, 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, ×, 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 In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts, ...

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'' {{Polyhedron navigator Prismatoid polyhedra Uniform polyhedra