Truncated dodecahedron

3D view of a truncated dodecahedron ( animation )

The truncated dodecahedron is a polyhedron ( polyhedron ) that is created by blunting the corners of a pentagon dodecahedron and is one of the Archimedean solids . Instead of the 20 corners of the dodecahedron there are now just as many equilateral triangles ; the pentagons of the dodecahedron become regular decagons .

The polyhedron that is dual to the truncated dodecahedron is the triacis icosahedron .

Formulas

Sizes of a truncated dodecahedron with edge length a
volume	${\displaystyle V={\frac {5}{12}}\,a^{3}\left(99+47{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=5a^{2}\left({\sqrt {3}}+6{\sqrt {5+2{\sqrt {5}}}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{4}}{\sqrt {74+30{\sqrt {5}}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{4}}\left(5+3{\sqrt {5}}\right)}$
1. Diagonal angle  ( decagon - decagon )  ≈ 116 ° 33 ′ 54 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {1}{5}}{\sqrt {5}}}$
2.  Diagonal angle (decagon - trigon )  ≈ 142 ° 37 ′ 21 ″	${\displaystyle \cos \,\alpha _{2}=-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}}$
3D edge angle  (decagon trigon)  ≈ 159 ° 5 ′ 41 ″	${\displaystyle \cos \,\beta =-{\frac {\sqrt {3}}{6}}\left(1+{\sqrt {5}}\right)}$
Corners solid angle  ≈ 1.2323 π	${\displaystyle \Omega =2\pi -\arccos \left(-{\frac {1}{3}}{\sqrt {5}}\right)}$

Web links

Commons : Truncated Dodecahedron  - collection of images, videos and audio files

Wiktionary: truncated dodecahedron  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Truncated Dodecahedron . In: MathWorld (English).