# Augmented pentagonal prism

Augmented pentagonal prism | |
---|---|

Type | JohnsonJ_{51} – – J_{52}J_{53} |

Faces | 4 triangles 4 squares 2 pentagons |

Edges | 19 |

Vertices | 11 |

Vertex configuration | 2+4(4^{2}.5)1(3 ^{4})4(3 ^{2}.4.5) |

Symmetry group | C_{2v} |

Properties | convex |

Net | |

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.

## Construction

[edit]The augmented pentagonal prism can be constructed from a pentagonal prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation.^{[1]} This square pyramid covers the square face of the prism, so the resulting polyhedron has four equilateral triangles, four 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 52nd Johnson solid .^{[3]}

## Properties

[edit]An augmented 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

[edit]**^**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

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