Prism (geometry)

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For other uses, see Prism (disambiguation).
Set of uniform prisms
Uniform prisms
(A hexagonal prism is shown)
Typeuniform polyhedron
Faces2+n total:
2 {n}
n {4}
Edges3n
Vertices2n
Schläfli symbol{n}×{} or t{2, n}
Coxeter-Dynkin diagramCDel node 1.pngCDel n.pngCDel node.pngCDel 2.pngCDel node 1.png
Vertex configuration4.4.n
Symmetry groupDnh, [n,2], (*n22), order 4n
Rotation groupDn, [n,2]+, (n22), order 2n
Dual polyhedronbipyramids
Propertiesconvex, semi-regular vertex-transitive
Generalized prisim net.svg
n-gonal prism net (n = 9 here)

In geometry, a prism is a polyhedron with an n-sided polygonal base, another congruent parallel base (with the same rotational orientation), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All cross-sections parallel to the base faces are congruent to the bases. Prisms are named for their base, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

General, right and uniform prisms[edit]

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. 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.

Some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. The term uniform prism can be used for a right prism with square sides, since such prisms are in the set of uniform polyhedra.

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

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.

The dual of a right prism is a bipyramid.

A parallelepiped is a prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms.

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.

Volume[edit]

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:

V = B \cdot h

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:

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

Surface area[edit]

The surface area 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.

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

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

Symmetry[edit]

The symmetry group of a right n-sided prism with regular base is Dnh of order 4n, except in the case of a cube, which has the larger symmetry group Oh of order 48, which has three versions of D4h as subgroups. The rotation group is Dn of order 2n, 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 Dnh contains inversion iff n is even.

Prismatic polytope[edit]

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 fi i-face elements (i = 0, ..., n). Its (n + 1)-polytope prism will have 2fi + fi−1 i-face elements. (With f−1 = 0, fn = 1.)

By dimension:

Uniform prismatic polytope[edit]

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:

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
Symmetry3456789101112
[2n,2]
[n,2]
[2n,2+]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 10.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 12.pngCDel node h.pngCDel 2.pngCDel node h.png
ImageTriangular prism.pngTetragonal prism.png
Uniform polyhedron 222-t012.png
Cube rotorotational symmetry.png
Pentagonal prism.pngHexagonal prism.png
Truncated triangle prism.png
Cantic snub hexagonal hosohedron.png
Prism 7.pngOctagonal prism.png
Truncated square prism.png
Cantic snub octagonal hosohedron.png
Prism 9.pngDecagonal prism.pngHendecagonal prism.pngDodecagonal prism.png
As spherical polyhedra
ImageSpherical triangular prism.pngSpherical square prism.png
Spherical square prism2.png
Spherical pentagonal prism.pngSpherical hexagonal prism.png
Spherical hexagonal prism2.png
Spherical heptagonal prism.pngSpherical octagonal prism.png
Spherical octagonal prism2.png
Spherical decagonal prism.png
Spherical decagonal prism2.png

See also[edit]

References[edit]

External links[edit]