Biaugmented pentagonal prism
Biaugmented pentagonal prism | |
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Type | Johnson J52 – J53 – J54 |
Faces | 8 equilateral triangles 3 squares 2 pentagons |
Edges | 23 |
Vertices | 12 |
Vertex configuration | 2(42.5) 2(34) 2x4(32.4.5) |
Symmetry group | C2v |
Properties | convex |
Net | |
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In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.
Construction
The biaugmented pentagonal prism can be constructed from a pentagonal prism by attaching two equilateral square pyramids to each of its square faces, a process known as augmentation.[1] These square pyramids cover the square face of the prism, so the resulting polyhedron has eight equilateral triangles, three squares, and two regular pentagons as its faces.[2] A convex polyhedron in which all faces are regular is Johnson solid, and the augmented pentagonal prism is among them, enumerated as 53rd Johnson solid .[3]
Properties
An biaugmented pentagonal prism with edge length has a surface area, calculated by adding the area of four equilateral triangles, four squares, and two regular pentagons:[2] Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:[2]
The dihedral angle of an augmented pentagonal prism can be calculated by adding the dihedral angle of an equilateral square pyramid and the regular pentagonal prism:[4]
- the dihedral angle of an augmented pentagonal prism between two adjacent triangular faces is that of an equilateral square pyramid between two adjacent triangular faces, ,
- the dihedral angle of an augmented pentagonal prism between two adjacent square faces is the internal angle of a regular pentagon .
- the dihedral angle of an augmented pentagonal prism between square-to-pentagon is that of a regular pentagonal prism between its base and its lateral faces .
- the dihedral angle of an augmented pentagonal prism between pentagon-to-triangle is , for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face , and the dihedral angle of a regular pentagonal prism between its base and its lateral face.
- the dihedral angle of an augmented pentagonal prism between square-to-triangle is , for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face, and the dihedral angle of a regular pentagonal prism between two adjacent squares.
References
- ^ Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
- ^ a b c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
- ^ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
- ^ 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. S2CID 122006114. Zbl 0132.14603.
External links
- Weisstein, Eric W. "Johnson Solid". MathWorld.
- Weisstein, Eric W. "Biaugmented pentagonal prism". MathWorld.
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- 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
- 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 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
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