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

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 have regular faces but are not uniform (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.[1]

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]

V=(\frac{1}{12}(\sqrt{2}+3\sqrt{3}))a^3\approx0.550864...a^3

A=(3+\sqrt{3})a^2\approx4.73205...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

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

Johnson solid No.7(Elongated triangular pyramid) fill the space with either or combination of (Octahedron,Johnson solid No.1(Square pyramid)).[3]

References

  1. ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
  2. ↑ Stephen Wolfram, "Elongated triangular pyramid" from Wolfram Alpha. Retrieved July 21, 2010.
  3. ↑ http://woodenpolyhedra.web.fc2.com/J7.html

External links