Icosidodecahedron

3D view of an icosidodecahedron ( animation )

The icosidodecahedron is a polyhedron ( polyhedron ) with 32 faces (12 pentagons and 20 equilateral triangles ), 30 corners and 60 edges of equal length.

It is described by the intersection of the penetration of a dodecahedron and icosahedron , which also appear in its name.

It is an Archimedean solid and dual to the rhombic triacontahedron .

Each 10 edges of the icosidodecahedron form the edges of a regular decagon . There are a total of six such independent, equilateral decagons in an icosidodecahedron.

Formulas

Sizes of an icosidodecahedron with edge length a
volume	${\displaystyle V={\frac {a^{3}}{6}}\left(45+17{\sqrt {5}}\right)}$
Surface area	${\displaystyle A_{O}=a^{2}\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{2}}\left(1+{\sqrt {5}}\right)}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}{\sqrt {5+2{\sqrt {5}}}}}$
Face angle  ≈ 142 ° 37 ′ 21 ″	${\displaystyle \cos \,\alpha =-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}}$
Corners solid angle  ≈ 1.1694 π	${\displaystyle \Omega =2\pi -\arccos \left({\frac {3+16{\sqrt {5}}}{-45}}\right)}$

Web links

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

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