Augmented dodecahedron

In geometry, the augmented dodecahedron is a Johnson solid combining a regular dodecahedron and a pentagonal pyramid.

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

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 ${\displaystyle J_{58}}$.^[2]

Properties

The surface area of an augmented dodecahedron ${\displaystyle A}$ is obtained by summing the area of its faces, eleven regular pentagons and five equilateral triangles. Its volume ${\displaystyle V}$ is obtained by adding the volume of a regular dodecahedron and a pentagonal pyramid, as suggested by the construction:^[1] $${\displaystyle {\begin{aligned}A&=11\cdot {\frac {\sqrt {25+10{\sqrt {5}}}}{4}}a^{2}+5\cdot {\frac {\sqrt {3}}{4}}a^{2}\approx 21.09a^{2},\\V&={\frac {15+7{\sqrt {5}}}{4}}a^{3}+{\frac {5+{\sqrt {5}}}{24}}a^{3}\approx 7.965a^{3}.\end{aligned}}}$$