Editing Snub dodecahedron (section)

==Metric properties==
For a snub dodecahedron whose edge length is 1, the surface area is
<math display=block>A = 20\sqrt{3} + 3\sqrt{25+10\sqrt{5}} \approx 55.286\,744\,958\,445\,15.</math>
Its volume is
<math display=block>V= \frac{(3\varphi+1)\xi^2+(3\varphi+1)\xi-\varphi/6-2}{\sqrt{3\xi^2-\varphi^2}}
 \approx 37.616\,649\,962\,733\,36.</math>
Alternatively, this volume may be written as 
<math display=block>\begin{align}
  V &= \frac{5+5\sqrt{5}}{6\sqrt{3}}\sqrt{{{18+6\sqrt{5}}} + {a \left({3+3\sqrt{5}} + a \right)} }
+\frac{5+3\sqrt{5}}{24\sqrt{2}}\sqrt{ 72 + {\left({5+\sqrt{5}}\right)}a \left({{3+3\sqrt{5}}} + a \right)} \\[2pt]
  &\approx 37.616\,649\,962\,733\,36,
\end{align}</math>
where
<math display=block>\begin{align}
  a &= \sqrt[3]{54(1+\sqrt{5})+6\sqrt{102+162\sqrt{5}}}+\sqrt[3]{54(1+\sqrt{5})-6\sqrt{102+162\sqrt{5}}} \\[2pt]
  &\approx 10.293\,368\,998\,184\,21.
\end{align}</math>
Its circumradius is
<math display=block>R = \frac12\sqrt{\frac{2-\xi}{1-\xi}} \approx 2.155\,837\,375.</math>
Its midradius is
<math display=block>r=\frac{1}{2}\sqrt{\frac{1}{1-\xi}}\approx 2.097\,053\,835\,25.</math>

There are two inscribed spheres, one touching the triangular faces, and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively:
<math display=block>\begin{align}
  r_3 &= \frac{\varphi\sqrt3}{6\xi}\sqrt{\frac{1}{1-\xi}}\approx 2.077\,089\,659\,74 \\[4pt]
  r_5 &= \frac12\sqrt{\varphi^2\xi^2 + 3\varphi^2\xi + \frac{11}{5}\varphi + \frac{12}{5}}\approx 1.980\,915\,947\,28.
\end{align}</math>Alternatively, ''r''<sub>3</sub>  can be expressed as a real root of this polynomial:

2985984 x<sup>12</sup> - 14183424 x<sup>10</sup> + 5723136 x<sup>8</sup> - 478656 x<sup>6</sup> + 12528 x<sup>4</sup> - 360 x<sup>2</sup> + 1 

The four positive real roots of the [[sextic]] equation in ''R''<sup>2</sup>
<math display="block">4096R^{12} - 27648R^{10} + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0</math>
are the circumradii of the snub dodecahedron (''U''<sub>29</sub>), [[great snub icosidodecahedron]] (''U''<sub>57</sub>), [[great inverted snub icosidodecahedron]] (''U''<sub>69</sub>), and [[great retrosnub icosidodecahedron]] (''U''<sub>74</sub>).

The snub dodecahedron has the highest [[sphericity]] of all Archimedean solids. If sphericity is defined as the ratio of volume squared over surface area cubed, multiplied by a constant of 36{{pi}} (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.<ref>{{citation
 | last = Aravind | first = P. K.
 | title = How Spherical Are the Archimedean Solids and Their Duals?
 | date = March 2011
 | journal = The College Mathematics Journal
 | volume = 42
 | issue = 2
 | pages = 98&ndash;107
 | doi = 10.4169/college.math.j.42.2.098
}}</ref>