# Pentagonal bipyramid

Pentagonal bipyramid
TypeBipyramid,
Johnson
J12J13J14
Faces10 triangles
Edges15
Vertices7
Vertex configurationV4.4.5
Schläfli symbol{ } + {5}
Coxeter diagram
Symmetry groupD5h, [5,2], (*225), order 20
Rotation groupD5, [5,2]+, (225), order 10
Dual polyhedronpentagonal prism
Propertiesconvex, face-transitive, (deltahedron)
Net
Johnson solid J13

In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (J13). Each bipyramid is the dual of a uniform prism.

Although it is face-transitive, it is not a Platonic solid because some vertices have four faces meeting and others have five faces.

## Properties

If the faces are equilateral triangles, it is a deltahedron and a Johnson solid (J13). It can be seen as two pentagonal pyramids (J2) connected by their bases.

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]

The pentagonal dipyramid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[2]

## 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 ${\displaystyle L}$:[3]

${\displaystyle H=L\cdot {\sqrt {2-{\frac {2}{\sqrt {5}}}}}\approx L\cdot 1.0514622242}$
${\displaystyle A=L^{2}\cdot {\frac {5{\sqrt {3}}}{2}}\approx L^{2}\cdot 4.330127019}$
${\displaystyle V=L^{3}\cdot {\frac {5+{\sqrt {5}}}{12}}\approx L^{3}\cdot 0.6030056648}$
Spherical pentagonal bipyramid

## Related polyhedra

The pentagonal bipyramid, dt{2,5}, can be in sequence rectified, rdt{2,5}, truncated, trdt{2,5} and alternated (snubbed), srdt{2,5}:

The dual of the Johnson solid pentagonal bipyramid is the pentagonal prism, with 7 faces: 5 rectangular faces and 2 pentagons.

Dual pentagonal bipyramid Net of dual

"Regular" right (symmetric) n-gonal bipyramids:
Bipyramid name Digonal bipyramid Triangular bipyramid
(See: J12)
Square bipyramid
(See: O)
Pentagonal bipyramid
(See: J13)
Hexagonal bipyramid Heptagonal bipyramid Octagonal bipyramid Enneagonal bipyramid Decagonal bipyramid ... Apeirogonal bipyramid
Polyhedron image ...
Spherical tiling image Plane tiling image
Face config. V2.4.4 V3.4.4 V4.4.4 V5.4.4 V6.4.4 V7.4.4 V8.4.4 V9.4.4 V10.4.4 ... V∞.4.4
Coxeter diagram ...

## 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. ^ Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010), "On well-covered triangulations. III", Discrete Applied Mathematics, 158 (8): 894–912, doi:10.1016/j.dam.2009.08.002, MR 2602814.
3. ^ Sapiña, R. "Area and volume of the Johnson solid J₁₃". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-09-04.