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 ...

, the pentagonal prism is a prism with a

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 ...

al 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 a

semiregular polyhedron 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 ...

, more generally, a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruence (geometry), congruent. Unifor ...

, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a '' truncated pentagonal hosohedron'', represented by

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 mo ...

t. Alternately it can be seen as the

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\t ...

of a regular pentagon and 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 i ...

, and represented by the product ×. The dual of a pentagonal prism is a

pentagonal bipyramid In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (). Each bipyramid is the dual of a uniform prism. Although it is face-transitive, it is not a Plato ...

. 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 pentagonal prism is ''D_5h'' of order 20. The rotation group is ''D₅'' of order 10.

Volume

The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism with edges ''h'' the formula is :

\frac\sqrt \approx 1.72h^3

Use

Nonuniform pentagonal prisms called

pentaprism A pentaprism is a five-sided reflecting prism used to deviate a beam of light by a constant 90°, even if the entry beam is not at 90° to the prism. The beam reflects inside the prism ''twice'', allowing the transmission of an image through a ...

s are also used in optics to rotate an image through a

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...

without changing its

chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

In 4-polytopes

It exists as cells of four nonprismatic uniform 4-polytopes in four dimensions:

Related polyhedra

External links

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Pentagonal Prism Polyhedron Model
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