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 triangular prism is a three-sided
prism; it 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 ...
made of a
triangular base, 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, and 3 faces joining
corresponding sides. A right triangular prism has
rectangular sides, otherwise it is ''oblique''. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.
Equivalently, it is a polyhedron of which two faces are parallel, while the
surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are
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. All cross-sections parallel to the base faces are the same triangle.
As a semiregular (or uniform) polyhedron
A right triangular prism is
semiregular or, more generally, a
uniform polyhedron if the base faces are equilateral
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s, and the other three faces are
squares. It can be seen as a
truncated trigonal hosohedron, represented by
Schläfli symbol t. Alternately it can be seen as the
Cartesian product of a triangle and a
line segment, and represented by the product, The
dual of a triangular prism is a
triangular bipyramid
In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
As the name suggests, ...
.
The
symmetry group of a right 3-sided prism with triangular base is
''D3h'' of order 12. The
rotation group is ''D
3'' of order 6. The symmetry group does not contain
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
* ...
.
Volume
The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to
compute the area of the triangle and multiply this by the length of the prism:
:
where is the length of one side of the triangle, is the length of an
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
drawn to that side, and is the distance between the triangular faces.
Truncated triangular prism
A ''truncated right triangular prism'' has one triangular face truncated (
planed) at an oblique angle.
:
The volume of a truncated triangular prism with base area ''A'' and the three heights ''h''
1, ''h''
2, and ''h''
3 is determined by
::
Facetings
There are two full D
3h symmetry
faceting
Stella octangula as a faceting of the cube
In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
New edges of a faceted polyhedron may be ...
s of a ''triangular prism'', both with 6
isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C
3v symmetry facetings have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.
Related polyhedra and tilings
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform
truncated polyhedra with
vertex configurations (3.2n.2n), and
,3 Coxeter group symmetry.
This polyhedron is topologically related as a part of sequence of
cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
. These
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
figures have (*n32) reflectional
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
.
Compounds
There are 4 uniform compounds of triangular prisms:
:
Compound of four triangular prisms,
compound of eight triangular prisms,
compound of ten triangular prisms,
compound of twenty triangular prisms.
Honeycombs
There are 9 uniform honeycombs that include triangular prism cells:
:
Gyroelongated alternated cubic honeycomb,
elongated alternated cubic honeycomb,
gyrated triangular prismatic honeycomb,
snub square prismatic honeycomb,
triangular prismatic honeycomb,
triangular-hexagonal prismatic honeycomb,
truncated hexagonal prismatic honeycomb
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms.
It is constructed from a triangular tiling extruded into pri ...
,
rhombitriangular-hexagonal prismatic honeycomb,
snub triangular-hexagonal prismatic honeycomb,
elongated triangular prismatic honeycomb
Related polytopes
The triangular prism is first in a dimensional series of
semiregular polytopes. Each progressive
uniform polytope is constructed
vertex figure of the previous polytope.
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, ...
identified this series in 1900 as containing all
regular polytope facets, containing all
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es and
orthoplexes (
equilateral triangles and
squares in the case of the triangular prism). In
Coxeter's notation the triangular prism is given the symbol −1
21.
Four dimensional space
The triangular prism exists as cells of a number of four-dimensional
uniform 4-polytopes, including:
See also
*
Wedge (geometry)
References
* {{mathworld , urlname = TriangularPrism , title = Triangular prism
Interactive Polyhedron: Triangular PrismSurface area and volume of a triangular prism
Prismatoid polyhedra
Space-filling polyhedra