Large rhombicosidodecahedron

3D view of a truncated icosidodecahedron ( animation )

The large rhombicosidodecahedron (also called truncated icosidodecahedron ) is a polyhedron that is one of the Archimedean solids . It is made up of 30 squares , 20 regular hexagons and 12 regular decagons and is dual to the hexakis icosahedron .

The name of the Great Rhombenikosidodekaeders based on the fact that the 30 squares congruent to the 30 diamonds of a circumscribed Rhombentriakontaeders are.

Formulas

Sizes of a truncated icosidodecahedron with edge length a
volume	${\displaystyle V=5a^{3}\left(19+10{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=30a^{2}\left(1+{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{2}}{\sqrt {31+12{\sqrt {5}}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}{\sqrt {30+12{\sqrt {5}}}}}$
1. Diagonal angle  ( hexagon square)  ≈ 159 ° 5 ′ 41 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {\sqrt {3}}{6}}\left(1+{\sqrt {5}}\right)}$
2. Diagonal angle  ( decagon square)  ≈ 148 ° 16 ′ 57 ″	${\displaystyle \cos \,\alpha _{2}=-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}$
3. Face angle  (decagon – hexagon)  ≈ 142 ° 37 ′ 21 ″	${\displaystyle \cos \,\alpha _{3}=-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}}$
Corners solid angle  = 1.5 π	${\displaystyle \Omega =\,2\pi -\arccos 0}$

Web links

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

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

• Eric W. Weisstein : Large rhombicosidodecahedron . In: MathWorld (English).