Augmented dodecahedron

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

File:J58 augmented dodecahedron.stl

3D model of an augmented dodecahedron

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.Template:R/superscript The augmented is Johnson solid, a convex polyhedron with regular faces, enumerated as the fifty-eighth $J_{58}$.Template:R/superscript

Properties

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

External links

• Weisstein, Eric W., "Augmented dodecahedron" ("Johnson Solid") at MathWorld.