Elongated triangular pyramid

Elongated triangular pyramid
Type	Johnson J6 – J7 – J8
Faces	1+3 triangles 3 squares
Edges	12
Vertices	7
Vertex configuration	1(33) 3(3.42) 3(32.42)
Symmetry group	C3v, [3], (*33)
Rotation group	C3, [3]+, (33)
Dual polyhedron	self
Properties	convex
Net

Johnson solid J₇.

In geometry, the elongated triangular pyramid is one of the Johnson solids (J₇). 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 topologically (but not geometrically) self-dual.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:

$V=\left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\approx 0.550864...a^{3}$

$A=\left(3+{\sqrt {3}}\right)a^{2}\approx 4.73205...a^{2}$

The height is given by

$H=a\cdot \left(1+{\frac {\sqrt {6}}{3}}\right)\approx a\cdot 1.816496581$

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.

Dual elongated triangular pyramid	Net of dual

Related polyhedra and honeycombs

The elongated triangular pyramid can form a tessellation of space with square pyramids and/or octahedra.

External links

• Weisstein, Eric W., "Johnson solid" ("Elongated triangular pyramid") at MathWorld.