Elongated triangular pyramid
Encyclopedia
In geometry
, the elongated triangular pyramid is one of the Johnson solid
s (J7). Norman Johnson discovered elongated triangular pyramids. As the name suggests, it can be constructed by elongating a tetrahedron
by attaching a triangular prism
to its base. Like any elongated pyramid
, the resulting solid is self-dual
.
The 92 Johnson solids were named and described by Norman Johnson in 1966.
e for volume
and surface area
can be used if all faces
are regular
, with edge length a:
If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the elongated triangular pyramid is one of the Johnson solid
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
s (J7). Norman Johnson discovered elongated triangular pyramids. As the name suggests, it can be constructed by elongating a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
by attaching a triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
to its base. Like any elongated pyramid
Pyramid
A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...
, the resulting solid is self-dual
Dual polyhedron
In 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...
.
The 92 Johnson solids were named and described by Norman Johnson in 1966.
Formulae
The following formulaFormula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
e for volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
and surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
can be used if all faces
Face (geometry)
In 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...
are regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
, with edge length a:
If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.
Dual polyhedron
The dual of the elongated triangular pyramid has 7 faces: 4 triangular, and 3 trapezoidal.| Dual elongated triangular pyramid | Net of dual |
|---|---|