In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
thumb|3D model of a truncated dodecahedron
Construction
The truncated dodecahedron is constructed from a regular dodecahedron by cutting all of its vertices off, a process known as truncation. Alternatively, the truncated dodecahedron can be constructed by expansion: pushing away the edges of a regular dodecahedron, forming the pentagonal faces into decagonal faces, as well as the vertices into triangles. Therefore, it has 32 faces, 90 edges, and 60 vertices.
The truncated dodecahedron may also be constructed by using Cartesian coordinates. With an edge length <math> 2\varphi - 2 </math> centered at the origin, they are all even permutations of
<math display="block">
\left(0, \pm \frac{1}{\varphi}, \pm (2 + \varphi) \right), \qquad
\left(\pm \frac{1}{\varphi}, \pm \varphi, \pm 2 \varphi \right), \qquad
\left(\pm \varphi, \pm 2, \pm (\varphi + 1) \right),
</math>
where <math display="inline"> \varphi = \frac{1 + \sqrt{5{2} </math> is the golden ratio.
Properties
The surface area <math> A </math> and the volume <math> V </math> of a truncated dodecahedron of edge length <math> a </math> are:
<math display="block"> \begin{align}
A &= 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5\right) a^2 &&\approx 100.991a^2 \\
V &= \frac{5}{12} \left(99+47\sqrt{5}\right) a^3 &&\approx 85.040a^3
\end{align}</math>
The dihedral angle of a truncated dodecahedron between two regular dodecahedral faces is 116.57°, and that between triangle-to-dodecahedron is 142.62°.
The truncated dodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.
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Further reading
External links
- Editable printable net of a truncated dodecahedron with interactive 3D view