Rhombicosidodecahedron

3D view of a rhombicosidodecahedron ( animation )

Rhombicosidodecahedron - colored in five colors

The (small) rhombicosidodecahedron is a polyhedron that is one of the Archimedean solids . It consists of 20 regular triangles , 30 squares and 12 regular pentagons .

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

Ten edges of the rhombicosidodecahedron form a regular decagon . There are a total of twelve such independent, equilateral decagons in a rhombicosidodecahedron.

The body that is dual to the rhombicosidodecahedron is the deltoidal hexacontahedron .

Formulas

Sizes of a rhombicosidodecahedron with edge length a
volume	${\displaystyle V={\frac {a^{3}}{3}}\left(60+29{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=a^{2}\left(30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{2}}{\sqrt {11+4{\sqrt {5}}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}{\sqrt {10+4{\sqrt {5}}}}}$
1. Diagonal angle  ( pentagon square)  ≈ 148 ° 16 ′ 57 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {1}{10}}{\sqrt {50+10{\sqrt {5}}}}}$
2. Face angle  (square - trine )  ≈ 159 ° 5 ′ 41 ″	${\displaystyle \cos \,\alpha _{2}=-{\frac {\sqrt {3}}{6}}(1+{\sqrt {5}})}$
Corners solid angle  ≈ 1.4153 π	${\displaystyle \Omega =2\pi -\arccos \left({\frac {5-4{\sqrt {5}}}{15}}\right)}$

Web links

Commons : Rhombenicosidodecahedron  - collection of images, videos and audio files

• Eric W. Weisstein : Rhombenikosidodekaeder . In: MathWorld (English).