Platonic solid
Angles

There are a number of angles 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

sin (θ 2) = cos (π q) sin (π p). {\displaystyle \sin \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{p}}\right)}}.}

This is sometimes more conveniently expressed in terms of the tangent by

tan (θ 2) = cos (π q) sin (π h). {\displaystyle \tan \left({\frac {\theta }{2}}\right)={\frac {\cos \left({\frac {\pi }{q}}\right)}{\sin \left({\frac {\pi }{h}}\right)}}.}

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 π. The defect, δ, at any vertex of the Platonic solids { p, q } is

δ = 2 π − q π (1 − 2 p). {\displaystyle \delta =2\pi -q\pi \left(1-{2 \over p}\right).}

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

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

Ω = q θ − (q − 2) π. {\displaystyle \Omega =q\theta -(q-2)\pi.\,}

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 π 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 steradians. The constant φ = 1 + √ 5 / 2 is the golden ratio.
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