Biaugmented pentagonal prism

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.

Biaugmented pentagonal prism
TypeJohnson
J52J53J54
Faces8 equilateral triangles
3 squares
2 pentagons
Edges23
Vertices12
Vertex configuration2(42.5)
2(34)
2x4(32.4.5)
Symmetry groupC2v
Propertiesconvex
Net

Construction

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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

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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

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  1. ^ 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.
  2. ^ 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.
  3. ^ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
  4. ^ 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.
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