Editing Truncated icosahedron (section)

== Properties ==
The surface area <math> A </math> and the volume <math> V </math> of the truncated icosahedron of edge length <math> a </math> are:{{r|berman}}
<math display=block>\begin{align}
A &= \left ( 20 \cdot \frac32\sqrt{3} + 12 \cdot \frac54\sqrt{ 1 + \frac{2}{\sqrt{5}}} \right) a^2 \approx 72.607a^2 \\
V &= \frac{125+43\sqrt{5}}{4} a^3 \approx 55.288a^3. 
\end{align}</math>
The [[sphericity]] of a polyhedron <math> \Psi </math> describes how closely a polyhedron resembles a [[sphere]]. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1. In the case of a truncated icosahedron, it is:{{r|berman}}
<math display="block"> \Psi = \frac{6\pi^{1/2} V}{A^{3/2}} \approx 0.9504. </math>

The [[dihedral angle]] of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°.{{r|johnson}}

The truncated icosahedron 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}} It has the same symmetry as the regular icosahedron, the [[icosahedral symmetry]], and it also has the property of [[vertex-transitivity]].{{r|kocakoca|cromwell}} The polygonal faces that meet for every vertex are one pentagon and two hexagons, and the [[vertex figure]] of a truncated icosahedron is <math> 5 \cdot 6^2 </math>. The truncated icosahedron's dual is [[pentakis dodecahedron]], a [[Catalan solid]],{{r|williams}} shares the same symmetry as the truncated icosahedron.{{r|holden}}