- Truncated dodecahedron
In

geometry , the**truncated dodecahedron**is anArchimedean solid . It has 12 regulardecagon al faces, 20 regular triangular faces, 60 vertices and 90 edges. __TOC__**Geometric relations**This

polyhedron can be formed from adodecahedron by truncating (cutting off) the corners so thepentagon faces becomedecagon s and the corners becometriangle s.It is part of a truncation process between a dodecahedron and icosahedron:

It shares its

vertex arrangement with three uniform star polyhedra:It is used in the

cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.**Area and volume**The area "A" and the

volume "V" of a truncated dodecahedron of edge length "a" are::$A\; =\; 5\; (sqrt\{3\}+6sqrt\{5+2sqrt\{5)\; a^2\; approx\; 100.99076a^2$:$V\; =\; frac\{5\}\{12\}\; (99+47sqrt\{5\})\; a^3\; approx\; 85.0396646a^3$**Cartesian coordinates**The following

Cartesian coordinates define the vertices of a truncateddodecahedron with edge length 2(τ-1), centered at the origin:: (0, ±1/τ, ±(2+τ)): (±(2+τ), 0, ±1/τ): (±1/τ, ±(2+τ), 0): (±1/τ, ±τ, ±2τ): (±2τ, ±1/τ, ±τ): (±τ, ±2τ, ±1/τ): (±τ, ±2, ±τ

^{2}): (±τ^{2}, ±τ, ±2): (±2, ±τ^{2}, ±τ)where τ = (1+√5)/2 is the

golden ratio (also written φ).**ee also***

*icosahedron

*icosidodecahedron

*truncated icosahedron **References*** (Section 3-9)

**External links***

* [*http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra*]

* [*http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra*] The Encyclopedia of Polyhedra

* [*http://www.lifeisastoryproblem.org/explore/net_trunc_dodecahedron.pdf Printable net of a Truncated Dodecahedron*] [*http://www.lifeisastoryproblem.org Life is a Story Problem.org*]

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