Editing Platonic solid (section)

== Geometric properties ==
=== Angles ===
There are a number of [[angle]]s associated with each Platonic solid. The [[dihedral angle]] is the interior angle between any two face planes. The dihedral angle, ''θ'', of the solid {''p'',''q''} is given by the formula

<math display="block">\sin(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/p)}.</math>

This is sometimes more conveniently expressed in terms of the [[tangent (trigonometric function)|tangent]] by

<math display="block">\tan(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/h)}.</math>

The quantity ''h'' (called the [[Coxeter number]]) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.

The [[angular deficiency]] at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2{{pi}}. The defect, ''δ'', at any vertex of the Platonic solids {''p'',''q''} is

<math display="block">\delta = 2\pi - q\pi\left(1 - {2 \over p}\right).</math>

By a theorem of Descartes, this is equal to 4{{pi}} divided by the number of vertices (i.e. the total defect at all vertices is 4{{pi}}).

The three-dimensional analog of a plane angle is a [[solid angle]]. The solid angle, ''Ω'', at the vertex of a Platonic solid is given in terms of the dihedral angle by

<math display="block">\Omega = q\theta - (q - 2)\pi.\,</math>

This follows from the [[spherical excess]] formula for a [[spherical polygon]] and the fact that the [[vertex figure]] of the polyhedron {''p'',''q''} is a regular ''q''-gon.

The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4{{pi}} steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.

The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in [[steradian]]s. The constant ''φ'' = {{sfrac|1 + {{sqrt|5}}|2}} is the [[golden ratio]].

{| class="wikitable" style="text-align:center"
! Polyhedron
! [[Dihedral angle|Dihedral <br/>angle]] <br/>(''θ'')
! tan&nbsp;{{sfrac|''θ''|2}}
! [[Defect (geometry)|Defect]] <br/>(''δ'')
! Vertex [[solid angle]] (''Ω'')
! Face <br/>solid <br/>angle
|-
| [[tetrahedron]] || 70.53° || <math>1 \over {\sqrt 2}</math> || <math>\pi</math>
| <math>\arccos\left(\frac{23}{27}\right) \quad \approx 0.551286</math>
| <math>\pi</math>
|-
| [[cube]] || 90° || <math>1</math> || <math>\pi \over 2</math>
| <math>\frac{\pi}{2} \quad \approx 1.57080</math>
| <math>2\pi \over 3</math>
|-
| [[octahedron]] || 109.47° || <math>\sqrt 2</math> || <math>{2\pi} \over 3</math>
| <math>4\arcsin\left({1 \over 3}\right) \quad \approx 1.35935</math>
| <math>\pi \over 2</math>
|-
| [[Regular dodecahedron|dodecahedron]] || 116.57° || <math>\varphi</math> || <math>\pi \over 5</math>
| <math>\pi - \arctan\left(\frac{2}{11}\right) \quad \approx 2.96174</math>
| <math>\pi \over 3</math>
|-
| [[Regular icosahedron|icosahedron]] || 138.19° || <math>\varphi^2</math> || <math>\pi \over 3</math>
| <math>2\pi - 5\arcsin\left({2\over 3}\right) \quad \approx 2.63455</math>
| <math>\pi \over 5</math>
|}

=== 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.

===Point in space===
For an arbitrary point in the space of a Platonic solid with circumradius ''R'', whose distances to the centroid of the Platonic solid and its ''n'' vertices are ''L'' and ''d<sub>i</sub>'' respectively, and

<math display="block">S^{(2m)}_{[n]}= \frac 1n\sum_{i=1}^n d_i^{2m}</math>,

we have<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

<math display="block">\begin{align}
S^{(2)}_{[4]} = S^{(2)}_{[6]} = S^{(2)}_{[8]}= S^{(2)}_{[12]}= S^{(2)}_{[20]} &= R^2+L^2, \\[4px]
S^{(4)}_{[4]} = S^{(4)}_{[6]} = S^{(4)}_{[8]}= S^{(4)}_{[12]}= S^{(4)}_{[20]} &= \left(R^2+L^2\right)^2 + \frac 43 R^2L^2, \\[4px]
S^{(6)}_{[6]} = S^{(6)}_{[8]} = S^{(6)}_{[12]}= S^{(6)}_{[20]}&= \left(R^2+L^2\right)^3 + 4R^2L^2 \left(R^2+L^2\right), \\[4px]
S^{(8)}_{[12]} = S^{(8)}_{[20]} &= \left(R^2+L^2\right)^4 + 8R^2L^2 \left(R^2+L^2\right)^2+\frac {16}{5} R^4L^4, \\[4px]
S^{(10)}_{[12]} = S^{(10)}_{[20]} &= \left(R^2+L^2\right)^5 +\frac {40}{3}R^2L^2\left(R^2+L^2\right)^3+16R^4L^4\left(R^2+L^2\right).
\end{align}</math>
For all five Platonic solids, we have<ref name= Mamuka />

<math display="block">S^{(4)}_{[n]}+\frac {16}{9}R^4= \left(S^{(2)}_{[n]}+ \frac 23R^2\right)^2.</math>

If ''d<sub>i</sub>'' are the distances from the ''n'' vertices of the Platonic solid to any point on its circumscribed sphere, then<ref name= Mamuka />

<math display="block">4\left(\sum_{i=1}^n d_i^2\right)^2=3n \sum_{i=1}^n d_i^4.</math>

===Rupert property===
A polyhedron ''P'' is said to have the [[Rupert property]] if a polyhedron of the same or larger size and the same shape as ''P'' can pass through a hole in ''P''.<ref name="AllFive">{{cite journal | first1=Richard P. | last1=Jerrard  |first2=John E. | last2=Wetzel | first3=Liping | last3 = Yuan | title = Platonic Passages | journal = Mathematics Magazine | date = April 2017 | volume = 90 | issue = 2 | pages = 87–98 | publisher = [[Mathematical Association of America]] | location = Washington, DC | doi = 10.4169/math.mag.90.2.87| s2cid=218542147 }}</ref>
All five Platonic solids have this property.<ref name="AllFive" /><ref>{{citation|last=Schrek|first= D. J. E.|title=Prince Rupert's problem and its extension by Pieter Nieuwland|journal=[[Scripta Mathematica]]|volume=16|year=1950|pages=73–80 and 261–267}}</ref><ref>{{citation | last = Scriba | first = Christoph J. | issue = 9 | journal = Praxis der Mathematik | language = de | mr = 0497615 | pages = 241–246 | title = Das Problem des Prinzen Ruprecht von der Pfalz | volume = 10 | year = 1968}}</ref>