Elongated pentagonal rotunda

Elongated pentagonal rotunda
Elongated pentagonal rotunda
Type	Johnson ; J20 - J21 - J22
Faces	2x5 triangles ; 2x5 squares ; 1+5 pentagons ; 1 decagon
Edges	55
Vertices	30
Vertex configuration	10(42.10); 10(3.42.5); 2.5(3.5.3.5)
Symmetry group	C5v
Dual polyhedron	-
Properties	convex
Net

Last updated August 26, 2021

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

Formulae

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

$V={\frac {1}{12}}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 14.612...a^{3}$

$A={\frac {1}{2}}\left(20+{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\approx 32.3472...a^{2}$

Dual polyhedron

The dual of the elongated pentagonal rotunda has 30 faces: 10 isosceles triangles, 10 rhombi, and 10 quadrilaterals.

Dual elongated pentagonal rotunda	Net of dual

Related Research Articles

In geometry, the elongated pentagonal pyramid is one of the Johnson solids (J₉). As the name suggests, it can be constructed by elongating a pentagonal pyramid (J₂) by attaching a pentagonal prism to its base.

In geometry, the pentagonal rotunda is one of the Johnson solids (J₆). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.

In geometry, the elongated square cupola is one of the Johnson solids (J₁₉). As the name suggests, it can be constructed by elongating a square cupola (J₄) by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" removed.

In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids (J₄₃). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron, by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda (J₄₂).

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (J₄₂). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda (J₃₄) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda (J₄₃).

In geometry, the pentagonal cupola is one of the Johnson solids (J₅). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J₉₂).

In geometry, the elongated triangular bipyramid or triakis triangular prism is one of the Johnson solids (J₁₄), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid (J₁₂) by inserting a triangular prism between its congruent halves.

In geometry, the elongated pentagonal cupola is one of the Johnson solids (J₂₀). As the name suggests, it can be constructed by elongating a pentagonal cupola (J₅) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J₃₈) with its "lid" removed.

In geometry, the gyrobifastigium is the 26th Johnson solid (J₂₆). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

In geometry, the pentagonal orthobicupola is one of the Johnson solids (J₃₀). As the name suggests, it can be constructed by joining two pentagonal cupolae (J₅) along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola (J₃₁).

In geometry, the pentagonal gyrobicupola is one of the Johnson solids (J₃₁). Like the pentagonal orthobicupola (J₃₀), it can be obtained by joining two pentagonal cupolae (J₅) along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (J₃₈). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola (J₃₀) by inserting a decagonal prism between its two congruent halves. Rotating one of the cupolae through 36 degrees before inserting the prism yields an elongated pentagonal gyrobicupola (J₃₉).

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J₃₉). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola (J₃₁) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae (J₅) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola (J₃₈).

In geometry, the elongated triangular cupola is one of the Johnson solids (J₁₈). As the name suggests, it can be constructed by elongating a triangular cupola (J₃) by attaching a hexagonal prism to its base.

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (J₃₂). As the name suggests, it can be constructed by joining a pentagonal cupola (J₅) and a pentagonal rotunda (J₆) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (J₃₃).

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (J₃₃). Like the pentagonal orthocupolarotunda (J₃₂), it can be constructed by joining a pentagonal cupola (J₅) and a pentagonal rotunda (J₆) along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (J₃₆). As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae (J₃). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (J₃₅).

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (J₄₁). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda (J₃₃) by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola (J₅) or the pentagonal rotunda (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda (J₄₀).

In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (J₄₀). As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda (J₃₂) by inserting a decagonal prism between its halves. Rotating either the cupola or the rotunda through 36 degrees before inserting the prism yields an elongated pentagonal gyrocupolarotunda (J₄₁).

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 pentagonal rotunda" from Wolfram Alpha. Retrieved July 22, 2010.

External links

Eric W. Weisstein, Elongated pentagonal rotunda ( Johnson solid ) at MathWorld.

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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 pentagonal rotunda" from Wolfram Alpha. Retrieved July 22, 2010.

[1]

[2]

v t e 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 pentagonal rotunda

Type	Johnson J₂₀ - J₂₁ - J₂₂
Faces	2x5 triangles 2x5 squares 1+5 pentagons 1 decagon
Edges	55
Vertices	30
Vertex configuration	10(4².10) 10(3.4².5) 2.5(3.5.3.5)
Symmetry group	C_5v
Dual polyhedron	-
Properties	convex
Net