Elongated square pyramid

In geometry, the elongated square pyramid is one of the Johnson solids (J₈). As the name suggests, it can be constructed by elongating a square pyramid (J₁) by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self-dual.

Elongated square pyramid
Type	Johnson J7 - J8 - J9
Faces	4 triangles 1+4 squares
Edges	16
Vertices	9
Vertex configuration	4(4^3) 1(3^4) 4(3^2.4^2)
Symmetry group	C4v, [4], (*44)
Rotation group	C4, [4]^+, (44)
Dual polyhedron	self
Properties	convex
Net

Johnson solid J₈.

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 the height (${\displaystyle H}$), surface area (${\displaystyle A}$) and volume (${\displaystyle V}$) can be used if all faces are regular, with edge length ${\displaystyle L}$:[2]

${\displaystyle H=L\cdot \left(1+{\frac {\sqrt {2}}{2}}\right)\approx L\cdot 1.707106781}$

${\displaystyle A=L^{2}\cdot \left(5+{\sqrt {3}}\right)\approx L^{2}\cdot 6.732050808}$

${\displaystyle V=L^{3}\left(1+{\frac {\sqrt {2}}{6}}\right)\approx L^{3}\cdot 1.23570226}$

Dual polyhedron

The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal.

Dual elongated square pyramid	Net of dual

Related polyhedra and honeycombs

The elongated square pyramid can form a tessellation of space with tetrahedra,[3] similar to a modified tetrahedral-octahedral honeycomb.

See also

• Elongated square bipyramid

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. Sapiña, R. "Area and volume of the Johnson solid J₈". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-08-28.
3. http://woodenpolyhedra.web.fc2.com/J8.html

External links

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