Elongated triangular pyramid

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.^[1]

Formulae[]

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

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^[3]

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.^[4]

References[]

^ 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.
^ Stephen Wolfram, "Elongated triangular pyramid" from Wolfram Alpha. Retrieved July 21, 2010.
^ Sapiña, R. "Area and volume of the Johnson solid J₇". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-08-12.
^ "J7 honeycomb".

External links[]

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

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[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.

[pye-3] Sapiña, R. "Area and volume of the Johnson solid J₇". Problemas y Ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-08-12.

[4] "J7 honeycomb".

[1]

[2]

[3]

[4]

v t Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)

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	C_3v, [3], (*33)
Rotation group	C₃, [3]⁺, (33)
Dual polyhedron	self
Properties	convex
Net