Editing Platonic solid (section)

=== Radii, area, and volume ===
Another virtue of regularity is that the Platonic solids all possess three concentric spheres:

* the [[circumscribed sphere]] that passes through all the vertices,
* the [[midsphere]] that is tangent to each edge at the midpoint of the edge, and
* the [[inscribed sphere]] that is tangent to each face at the center of the face.

The [[radius|radii]] of these spheres are called the ''circumradius'', the ''midradius'', and the ''inradius''. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius ''R'' and the inradius ''r'' of the solid {''p'',&nbsp;''q''} with edge length ''a'' are given by

<math display="block">\begin{align}
  R &= \frac{a}{2} \tan\left(\frac{\pi}{q}\right)\tan\left(\frac{\theta}{2}\right) \\[3pt]
  r &= \frac{a}{2} \cot\left(\frac{\pi}{p}\right)\tan\left(\frac{\theta}{2}\right)
\end{align}</math>

where ''θ'' is the dihedral angle. The midradius ''ρ'' is given by

<math display="block">\rho = \frac{a}{2} \cos\left(\frac{\pi}{p}\right)\,{\csc}\biggl(\frac{\pi}{h}\biggr)</math>

where ''h'' is the quantity used above in the definition of the dihedral angle (''h'' = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in ''p'' and ''q'':

<math display="block">\frac{R}{r} =
  \tan\left(\frac{\pi}{p}\right) \tan\left(\frac{\pi}{q}\right) =
  \frac{{\sqrt{{\csc^{2}}\Bigl(\frac\theta2\Bigr) - {\cos^{2}}\Bigl(\frac\alpha2\Bigr)}}}{\sin\Bigl(\frac{\alpha}{2}\Bigr)}.
</math>

The [[surface area]], ''A'', of a Platonic solid {''p'',&nbsp;''q''} is easily computed as area of a regular ''p''-gon times the number of faces ''F''. This is:

<math display="block">A = \biggl(\frac{a}{2}\biggr)^2 Fp\cot\left(\frac{\pi}{p}\right).</math>

The [[volume]] is computed as ''F'' times the volume of the [[pyramid (geometry)|pyramid]] whose base is a regular ''p''-gon and whose height is the inradius ''r''. That is,

<math display="block">V = \frac{1}{3} rA.</math>

The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, ''a'', to be equal to 2.

{| class="wikitable" style="text-align:center"
|-
! rowspan=2 | Polyhedron, <br/>''a''&nbsp;=&nbsp;2
! colspan=3 | Radius
! rowspan=2 | Surface area, <br/>''A''
! colspan=2 | Volume
|-
! In-, ''r''
! Mid-, ''ρ''
! Circum-, ''R''
! ''V''
! Unit edges
|-
| [[tetrahedron]] || <math>1\over {\sqrt 6}</math> || <math>1\over {\sqrt 2}</math> || <math>\sqrt{3\over 2}</math> || <math>4\sqrt 3</math> || <math>\frac{\sqrt 8}{3}\approx 0.942809</math> || <math>\approx0.117851</math>
|- align=center
| [[cube]] || <math>1\,</math> || <math>\sqrt 2</math> || <math>\sqrt 3</math> || <math>24\,</math> || <math>8\,</math> || <math>1\,</math>
|-
| [[octahedron]] || <math>\sqrt{2\over 3}</math> || <math>1\,</math> || <math>\sqrt 2</math> || <math>8\sqrt 3</math> || <math>\frac{\sqrt {128}}{3}\approx 3.771236</math> || <math>\approx 0.471404</math>
|-
| [[regular dodecahedron|dodecahedron]] || <math>\frac{\varphi^2}{\xi}</math> || <math>\varphi^2</math> || <math>\sqrt 3\,\varphi</math> || <math>12 {\sqrt {25+10\sqrt5}}</math> || <math>\frac{20\varphi^3}{\xi^2}\approx 61.304952</math> || <math>\approx 7.663119</math>
|-
| [[icosahedron]] || <math>\frac{\varphi^2}{\sqrt 3}</math> || <math>\varphi</math> || <math>\xi\varphi</math> || <math>20\sqrt 3</math> || <math>\frac{20\varphi^2}{3}\approx 17.453560</math> || <math>\approx 2.181695</math>
|}

The constants ''φ'' and ''ξ'' in the above are given by

<math display="block">
  \varphi = 2\cos{\pi\over 5} = \frac{1+\sqrt 5}{2},\qquad
  \xi = 2\sin{\pi\over 5} = \sqrt{\frac{5-\sqrt 5}{2}} = \sqrt{3 - \varphi}.
</math>

Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume). The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.