Pentagonal rotunda

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Pentagonal rotunda
Pentagonal rotunda.png
Type Johnson
J5 - J6 - J7
Faces 10 triangles
1+5 pentagons
1 decagon
Edges 35
Vertices 20
Vertex configuration 2.5(3.5.3.5)
10(3.5.10)
Symmetry group C5v
Rotation group C5, [5]+, (55)
Dual polyhedron -
Properties convex
Net
Pentagonal Rotunda Net.svg

In geometry, the pentagonal rotunda is one of the Johnson solids (J6). It can be seen as half an icosidodecahedron.

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[edit]

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

V=\left(\frac{1}{12}\left(45+17\sqrt{5}\right)\right)a^3\approx6.91776...a^3

A=\left(\frac{1}{2}\left(5\sqrt{3}+\sqrt{10\left(65+29\sqrt{5}\right)}\right)\right)a^2=\left(\frac{1}{2}\sqrt{5\left(145+58\sqrt{5}+2\sqrt{30\left(65+29\sqrt{5}\right)}\right)}\right)a^2\approx22.3472...a^2

R=\left(\frac{1}{2}\left(1+\sqrt{5}\right)\right)a\approx1.61803...a

H=\left(\sqrt{1+\frac{2}{\sqrt{5}}}\right)a\approx1.37638...a

Dual polyhedron[edit]

The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.

Dual pentagonal rotunda Net of dual
Dual pentagonal rotunda.png Dual pentagonal rotunda net.png

References[edit]

  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, "Pentagonal Rotunda" from Wolfram Alpha. Retrieved July 21, 2010.

External links[edit]