Editing Platonic solid (section)

=== 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>
|}