53rd Johnson solid
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]
![{\displaystyle {\frac {6+4{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}}{2}}a^{2}\approx 9.9051a^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc23c453ad429b25abc4d56f7e66f38f0ecaffb6)
Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:
[2]
![{\displaystyle {\frac {\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}{12}}a^{3}\approx 2.1919a^{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b68227d6c05a7486e54f5d1905ea1fc23fd79008)
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
External links