In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, the
pentagonal prism is a
prismIn geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
with a
pentagonIn geometry, a pentagon is any fivesided polygon. A pentagon may be simple or selfintersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a selfintersecting pentagon. Regular pentagons :In a regular pentagon, all sides are equal in length and...
al base. It is a type of
heptahedronA heptahedron is a polyhedron having seven sides, or faces.A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven equilateral triangle faces...
with 7
facesIn geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
, 15
edgesIn geometry, an edge is a onedimensional line segment joining two adjacent zerodimensional vertices in a polygon. Thus applied, an edge is a connector for a onedimensional line segment and two zerodimensional objects....
, and 10
verticesIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.Of an angle:...
.
As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is a
semiregular polyhedronThe term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
, more generally, a
uniform polyhedronA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive...
, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a
truncatedIn geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. Uniform truncation :...
pentagonal hosohedron, represented by
Schläfli symbol t{2,5}. Alternately it can be seen as the
Cartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of a regular pentagon and a
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
, and represented by the product {5}x{}. The
dualIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
of a pentagonal prism is a pentagonal bipyramid.
The
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of a right pentagonal prism is
D_{5h}In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
of order 20. The
rotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of threedimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
is
D_{5} of order 10.
Volume
As in most prisms, the volume is found by taking the area of the base, with a side length of
a, and multiplying it by the height
h.
See also
 Set of prisms
In geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
 Triangular prism
In geometry, a triangular prism is a threesided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
 Cube
In geometry, a cube is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
Squarecapped prism
 Hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
Use
Nonuniform pentagonal prisms called
PentaprismA pentaprism is a fivesided reflecting prism used to deviate a beam of light by 90°. The beam reflects inside the prism twice, allowing the transmission of an image through a right angle without inverting it as an ordinary rightangle prism or mirror would.The reflections inside the prism are not...
s are also used in optics to rotate an image through a
right angleIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
without changing its
chiralityChirality is a property of asymmetry important in several branches of science. It may refer to:* Chirality , a property of molecules having a nonsuperimposable mirror image...
.
In polychora
It exists as cells of four
polychoraIn geometry, a uniform polychoron is a polychoron or 4polytope which is vertextransitive and whose cells are uniform polyhedra....
in 4 dimensions:
cantellated 600cell

cantitruncated 600cell

runcinated 600cell

runcitruncated 600cell





External links