In geometry, the augmented dodecahedron is a Johnson solid combining a regular dodecahedron and a pentagonal pyramid.
thumb|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. The augmented is Johnson solid, a convex polyhedron with regular faces, enumerated as the fifty-eighth <math> J_{58} </math>.
Properties
The surface area of an augmented dodecahedron <math> A </math> is obtained by summing the area of its faces, eleven regular pentagons and five equilateral triangles. Its volume <math> V </math> is obtained by adding the volume of a regular dodecahedron and a pentagonal pyramid, as suggested by the construction:
<math display="block">
\begin{align}
A &= 11 \cdot \frac{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{align} </math>
References
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