Elongated pentagonal pyramid

Elongated pentagonal pyramid
Elongated pentagonal pyramid
Type	Johnson ; J8 – J9 – J10
Faces	5 triangles ; 5 squares ; 1 pentagon
Edges	20
Vertices	11
Vertex configuration	5(42.5); 5(32.42); 1(35)
Symmetry group	C5v, [5], (*55)
Rotation group	C5, [5]+, (55)
Dual polyhedron	self
Properties	convex
Net

Last updated July 29, 2024

In geometry, the elongated pentagonal pyramid is one of the Johnson solids ( $J 9$ ). As the name suggests, it can be constructed by elongating a pentagonal pyramid ( $J 2$ ) by attaching a pentagonal prism to its base.

Formulae

The following formulae for the height ( $H$ ), surface area ( $A$ ) and volume ( $V$ ) can be used if all faces are regular, with edge length $L$ :^[2]

H=L\cdot \left(1+{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx L\cdot 1.525731112

A=L^{2}\cdot {\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\approx L^{2}\cdot 8.88554091

V=L^{3}\cdot \left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)\approx L^{3}\cdot 2.021980233

Dual polyhedron

The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.

Dual elongated pentagonal pyramid	Net of dual

Related Research Articles

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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.

In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.

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

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In geometry, the elongated triangular bipyramid or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.

In geometry, the elongated pentagonal bipyramid is a polyhedron constructed by attaching two pentagonal pyramids onto the base of a pentagonal prism. It is an example of Johnson solid.

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

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J₂₄). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J₅) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J₄₆) with one pentagonal cupola removed.

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal orthobicupola 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.

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In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.

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In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids. Like the pentagonal orthocupolarotunda, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda 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 orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

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

In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda 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.

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 .
↑ Sapiña, R. "Area and volume of the Johnson solid J₉". Problemas y ecuaciones (in Spanish). ISSN 2659-9899 . Retrieved 2020-08-30.

External links

Weisstein, Eric W., "Johnson solid" ("Elongated pentagonal 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 .

[pye-2] Sapiña, R. "Area and volume of the Johnson solid J₉". Problemas y ecuaciones (in Spanish). ISSN 2659-9899 . Retrieved 2020-08-30.

[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 pyramid

Type	Johnson $J 8 - J 9 - J 10$
Faces	5 triangles 5 squares 1 pentagon
Edges	20
Vertices	11
Vertex configuration	$5(4 2 .5) 5(3 2 .4 2) 1(3 5)$
Symmetry group	$C 5v, [5], (*55)$
Rotation group	$C 5, [5] +, (55)$
Dual polyhedron	self
Properties	convex
Net