Elongated pentagonal pyramid | |
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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 | |
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.
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 following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length : [2]
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 |
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In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid. Each bipyramid is the dual of a uniform prism.
In geometry, the triangular cupola is one of the Johnson solids. It can be seen as half a cuboctahedron.
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids. It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.
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 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 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. 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 elongated triangular pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically self-dual.
In geometry, the elongated square pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base. Like any elongated pyramid, it is topologically self-dual.
In geometry, the elongated triangular bipyramid or triakis triangular prism is one of the Johnson solids, convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid by inserting a triangular prism between its congruent halves.
In geometry, the elongated square bipyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating an octahedron by inserting a cube 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 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 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).