![Pentagonal prism](https://upload.wikimedia.org/wikipedia/commons/2/2a/Pentagonal_prism.stl)
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 ...
, the pentagonal prism is a
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
with a
pentagonal base. It is a type of
heptahedron with seven
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 ...
, 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 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 ...
, 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 t. Alternately it can be seen as the
Cartesian product of a regular pentagon and a
line segment, 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 Plat ...
.
The
symmetry group of a right pentagonal prism is
''D5h'' of order 20. 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
5'' 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
:
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 r ...
s are also used in optics to rotate an image through a
right angle without changing its
chirality.
In 4-polytopes
It exists as cells of four nonprismatic
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. Th ...
s in four dimensions:
Related polyhedra
:
External links
*
Pentagonal Prism Polyhedron Model-- works in your web browser
Prismatoid polyhedra
{{Polyhedron-stub