Augmented dodecahedron
| Augmented dodecahedron | |
|---|---|
 | |
| Type | Johnson J57 – J58 – J59 |
| Faces | 5 triangles 11 pentagons |
| Edges | 35 |
| Vertices | 21 |
| Vertex configuration | 3.5(53) 5(32.52) 1(35) |
| Symmetry group | C5v |
| Properties | convex |
| Net | |
 | |
In geometry, the augmented dodecahedron is a Johnson solid combining a regular dodecahedron and a pentagonal pyramid. File:J58 augmented dodecahedron.stl
Construction
An augmented dodecahedron is constructed from a regular dodecahedron, a twelve-sided polyhedron with regular pentagons, by attaching a regular-faced pentagonal pyramid to one of the regular dodecahedron's faces; the regular polygons mean that all of its internal angles and edges are equal. The resulting polyhedron covers one pentagon from a dodecahedron with five equilateral triangles from the pyramid. Ergo, the augmented dodecahedron has eleven pentagonal faces and five equilateral triangular faces, totaling sixteen faces.[1] The augmented is Johnson solid, a convex polyhedron with regular faces, enumerated as the fifty-eighth .[2]
Properties
The surface area of an augmented dodecahedron is obtained by summing the area of its faces, eleven regular pentagons and five equilateral triangles. Its volume is obtained by adding the volume of a regular dodecahedron and a pentagonal pyramid, as suggested by the construction:[1]
References
- ↑ 1.0 1.1 Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8.
- ↑ Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways 46 (3): 177. https://digitalcommons.butler.edu/wordways/vol46/iss3/9/.
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