- Icosidodecahedron
An

**icosidodecahedron**is apolyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of theArchimedean solid s and more particularly, aquasiregular polyhedron .An icosidodecahedron has icosahedral symmetry, and its first

stellation is the compound of adodecahedron and its dualicosahedron , with the vertices of the icosahedron located at the midpoints of the edges of either. Canonical coordinates for the vertices of an icosidodecahedron with unit edges are thecyclic permutation s of (0,0,±τ), (±1/2, ±τ/2, ±(1+τ)/2), where τ is thegolden ratio , (1+√5)/2. Itsdual polyhedron is therhombic triacontahedron . An icosidodecahedron can be split along several planes to formpentagonal rotunda e, which belong among theJohnson solid s.In the standard nomenclature used for the

Johnson solid s, an icosidodecahedron would be called a "pentagonal gyrobirotunda".**Area and volume**The area "A" and the volume "V" of the icosidodecahedron of edge length "a" are::$A\; =\; (5sqrt\{3\}+3sqrt\{25+10sqrt\{5)\; a^2\; approx\; 29.3059828a^2$:$V\; =\; frac\{1\}\{6\}\; (45+17sqrt\{5\})\; a^3\; approx\; 13.8355259a^3.$

**Related polyhedra**The icosidodecahedron is a rectified

dodecahedron and also a rectifiedicosahedron , existing as the full-edge truncation between these regular solids.The Icosidodecahedron contains 12 pentagons of the

dodecahedron and 20 triangles of theicosahedron :There are also 9 uniform star polyhedra which share the same

vertex arrangement :**See also***

Cuboctahedron

*Dodecahedron

*Great truncated icosidodecahedron

*Icosahedron

*Rhombicosidodecahedron

*Truncated icosidodecahedron **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_icosidodecahedron.pdf Printable Geometric Net of an Icosidodecahedron*] [*http://www.lifeisastoryproblem.org Life is a Story Problem.org*]

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**icosidodecahedron**— noun A semiregular polyhedron with twelve faces that are regular pentagons and twenty that are equilateral triangles … Wiktionary**icosidodecahedron**— … Useful english dictionary**Truncated icosidodecahedron**— (Click here for rotating model) Type Archimedean solid Uniform polyhedron Elements F = 62, E = 180, V = 120 (χ = 2) Faces by sides … Wikipedia**Great inverted snub icosidodecahedron**— Type Uniform star polyhedron Elements F = 92, E = 150 V = 60 (χ = 2) Faces by sides (20+60){3}+12{5/2} … Wikipedia**Small complex icosidodecahedron**— Type Uniform star polyhedron Elements F = 32, E = 60 (30x2) V = 12 (χ = 16) Faces by sides 20{3}+12{5} Wythoff sy … Wikipedia**Great icosidodecahedron**— In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.This polyhedron can be considered a rectified great icosahedron.It shares the same vertex arrangement with the icosidodecahedron.Unlike the great… … Wikipedia**Great ditrigonal icosidodecahedron**— In geometry, the great ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U47.It shares the vertex arrangement with the regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal… … Wikipedia**Small ditrigonal icosidodecahedron**— In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.It shares the vertex arrangement with the regular dodecahedron. It additionally shares its edge arrangement with the great ditrigonal… … Wikipedia**Great truncated icosidodecahedron**— In geometry, the great truncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U68.Cartesian coordinates Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even… … Wikipedia**Great snub icosidodecahedron**— In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57.This polyhedron can be considered a snub great icosahedron. Cartesian coordinates Cartesian coordinates for the vertices of a great snub… … Wikipedia