Elongated triangular pyramid

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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

In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). Norman Johnson discovered elongated triangular pyramids.[citation needed] 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 regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[1]

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

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

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

References

  1. ↑ Stephen Wolfram, "Elongated triangular pyramid" from Wolfram Alpha. Retrieved July 21, 2010.

External links


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