Editing Icosidodecahedron (section)

== Properties ==
The surface area of an icosidodecahedron {{mvar|A}} can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron {{mvar|V}} can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:{{r|berman}}
<math display="block">\begin{align}
A &= \left(5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right) a^2 &\approx 29.306a^2 \\
V &= \frac{45+17\sqrt{5}}{6}a^3 &\approx 13.836a^3.
\end{align}</math>

The [[dihedral angle]] of an icosidodecahedron between pentagon-to-triangle is
<math display="block"> \arccos \left(-\sqrt{\frac{5 + 2\sqrt{5}}{15}} \right) \approx 142.62^\circ, </math>
determined by calculating the angle of a pentagonal rotunda.{{r|williams}}

An icosidodecahedron has [[icosahedral symmetry]], and its first [[stellation]] is the [[Compound of dodecahedron and icosahedron|compound]] of a [[dodecahedron]] and its dual [[icosahedron]], with the vertices of the icosidodecahedron located at the midpoints of the edges of either.

The icosidodecahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.{{r|diudea}} The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the [[vertex figure]] of an icosidodecahedron is <math> (3 \cdot 5)^2 = 3^2 \cdot 5^2 </math>. Its [[dual polyhedron]] is [[rhombic triacontahedron]], a [[Catalan solid]].{{r|williams}}

{{multiple image
 | align = right  | total_width = 400
 | image1 = Polyhedron 12-20, davinci.png
 | image2 = Spherical icosidodecahedron with colored cicles.png
 | footer = The 60 edges form 6 [[decagon]]s corresponding to [[great circle]]s in the spherical tiling.
}}
The icosidodecahedron has 6 central [[decagon]]s. Projected into a sphere, they define 6 [[great circle]]s. {{harvtxt|Fuller|1975}} used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his [[31 great circles of the spherical icosahedron]].{{r|fuller}}

The long radius (center to vertex) of the icosidodecahedron is in the [[golden ratio]] to its edge length; thus its radius is {{mvar|φ}} if its edge length is 1, and its edge length is {{math|{{sfrac|1|''φ''}}}} if its radius is 1.{{r|williams}} Only a few uniform polytopes have this property, including the four-dimensional [[600-cell]], the three-dimensional icosidodecahedron, and the two-dimensional [[Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.) These ''radially golden'' polytopes can be constructed, with their radii, from [[Golden triangle (mathematics)|golden triangle]]s which meet at the center, each contributing two radii and an edge.