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 ...
, the elongated triangular pyramid is one of the
Johnson solids (). As the name suggests, it can be constructed by elongating a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
by attaching a
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. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...
to its base. Like any elongated
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
, the resulting solid is topologically (but not geometrically) self-
dual.
Formulae
The following
formulae for
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
and
surface area can be used if all
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 ...
are
regular, with edge length ''a'':
:
:
The
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
is given by
:
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
Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.
Related polyhedra and honeycombs
The elongated triangular pyramid can form a
tessellation of space
In geometry, a honeycomb is a ''space filling'' or '' close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dim ...
with
square pyramids and/or
octahedra
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
.
References
External links
*
Johnson solids
Self-dual polyhedra
Pyramids and bipyramids
{{Polyhedron-stub