In
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 o ...
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, 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 eq ...
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 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 ...
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, postu ...
. 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 c ...
'' to the base faces.
Example: a
parallelepiped is an oblique prism whose base is a
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 eq ...
, 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 ''
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 c ...
'' 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 bic ...
all the joining faces are ''
rectangular''.
The
dual 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 ...
.
A right prism (with rectangular sides) with
regular ''n''-gon bases has Schläfli symbol ×. It approaches a
cylindrical 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 am ...
; 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 ...
is considered a circular prism.
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 ××.
*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
squares.
Thus all the faces of a uniform prism are regular polygons. Also, such prisms are
isogonal; thus they are
uniform polyhedra alright. 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.
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). ...
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 or planar lamina, while '' surface area'' refers to the area of an op ...
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:
:
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:
:
Surface area
The surface
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 or planar lamina, while '' surface area'' refers to the area of an op ...
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
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 ...
with side length ''s'', and with height ''h'', is therefore:
:
Schlegel diagrams
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
Oh 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
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 ''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
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 o ...
top and bottom faces.
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 into tetrahedra without adding new vertices. The smallest case: the triangular form, 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'',2sup>+, order 2''n''. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.
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
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 operatio ...
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 × , 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 san ...
. 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 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 an ...
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
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 ...
. 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 one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
.
Prismatic polytope
A ''prismatic
polytope'' 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
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 ...
with ''n'' vertices, ''n'' edges. Its prism has 2''n'' vertices, 3''n'' edges, and faces.
*Take 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 o ...
with ''v'' vertices, ''e'' edges, and ''f'' faces. Its prism has 2''v'' vertices, edges, faces, and cells.
*Take a
polychoron 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
Schläfli symbol can form a uniform prismatic ()-polytope represented by a
Cartesian product of
two Schläfli symbols:
By dimension:
*A 0-polytopic prism is a
line segment, represented by an empty
Schläfli symbol .
*:
*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 contain ...
, made from 2 translated line segments. It is represented as the product Schläfli symbol ×. If it is
square, symmetry can be reduced:
*:Example: , Square, ×, two parallel line segments, connected by two line segment ''sides''.
*A
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 ...
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:
*:Example: ,
Pentagonal prism, ×, 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 sim ...
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: × =
*:Example: ,
Dodecahedral prism, ×, 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 ×, 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
tesseract, as is ×, a ''cubic prism''. ×× (4-4 duoprism prism), × (cube-4 duoprism) and × (tesseractic prism) are lower symmetry forms of a
5-cube.
See also
*
Apeirogonal prism
*
Rectified prism
In geometry, a rectified prism (also rectified bipyramid) is one of an infinite set of polyhedra, constructed as a rectification (geometry), rectification of an ''n''-gonal prism (geometry), prism, truncating the vertices down to the midpoint of th ...
*
Prismanes
*
List of shapes
Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
Mathematics
* List of mathematical shapes
* List of two- ...
References
* Chapter 2: Archimedean polyhedra, prisma and antiprisms
External links
*
Paper models of prisms and antiprismsFree nets of prisms and antiprisms
Paper models of prisms and antiprismsUsing nets generated by ''
Stella''
{{Polyhedron navigator
Prismatoid polyhedra
Uniform polyhedra