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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 elongated triangular pyramid is one of the
Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...
s (). 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 the ...
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''. ...
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, quadrila ...
, 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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
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 aff ...
are
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
, with edge length ''a'': :V=\left(\frac\left(\sqrt+3\sqrt\right)\right)a^3\approx0.550864...a^3 :A=\left(3+\sqrt\right)a^2\approx4.73205...a^2 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 :H = a\cdot \left( 1 + \frac\right) \approx a\cdot 1.816496581 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 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 o ...
with
square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid ...
s 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 at ...
.


References


External links

* Johnson solids Self-dual polyhedra Pyramids and bipyramids {{Polyhedron-stub