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

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 J 58 {\displaystyle J_{58}}.

Properties

The surface area of an augmented dodecahedron A {\displaystyle A} is obtained by summing the area of its faces, eleven regular pentagons and five equilateral triangles. Its volume V {\displaystyle V} is obtained by adding the volume of a regular dodecahedron and a pentagonal pyramid, as suggested by the construction: A = 11 ⋅ 25 + 10 5 4 a 2 + 5 ⋅ 3 4 a 2 ≈ 21.09 a 2 , V = 15 + 7 5 4 a 3 + 5 + 5 24 a 3 ≈ 7.965 a 3 . {\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}}}

External links