Regular dodecahedron
Relation to the golden ratio

The golden ratio is the ratio between two numbers equal to the ratio of their sum to the larger of the two quantities. It is one of two roots of a polynomial, expressed as ϕ = 1 + 5 2 ≈ 1.618 {\textstyle \phi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.618}. The golden ratio can be applied to the regular dodecahedron's metric properties, as well as to construct the regular dodecahedron.

The surface area A {\displaystyle A} and the volume V {\displaystyle V} of a regular dodecahedron of edge length a {\displaystyle a} are: A = 15 ϕ 3 − ϕ a 2, V = 5 ϕ 3 6 − 2 ϕ a 3. {\displaystyle A={\frac {15\phi }{\sqrt {3-\phi }}}a^{2},\qquad V={\frac {5\phi ^{3}}{6-2\phi }}a^{3}.}

The following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented: (± 1, ± 1, ± 1), (0, ± ϕ, ± 1 / ϕ), (± 1 / ϕ, 0, ± ϕ), (± ϕ, ± 1 / ϕ, 0). {\displaystyle {\begin{aligned}(\pm 1,\pm 1,\pm 1),&\qquad (0,\pm \phi,\pm 1/\phi),\\(\pm 1/\phi,0,\pm \phi),&\qquad (\pm \phi,\pm 1/\phi,0).\end{aligned}}}

If the edge length of a regular dodecahedron is a {\displaystyle a}, the radius of a circumscribed sphere r u {\displaystyle r_{u}} (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere r i {\displaystyle r_{i}} (tangent to each of the regular dodecahedron's faces), and the midradius r m {\displaystyle r_{m}} (one that touches the middle of each edge) are: r u = ϕ 3 2 a ≈ 1.401 a, r i = ϕ 2 2 3 − ϕ a ≈ 1.114 a, r m = ϕ 2 2 a ≈ 1.309 a. {\displaystyle {\begin{aligned}r_{u}&={\frac {\phi {\sqrt {3}}}{2}}a\approx 1.401a,\\r_{i}&={\frac {\phi ^{2}}{2{\sqrt {3-\phi }}}}a\approx 1.114a,\\r_{m}&={\frac {\phi ^{2}}{2}}a\approx 1.309a.\end{aligned}}} Note that, given a regular dodecahedron of edge length one, r u {\displaystyle r_{u}} is the radius of a circumscribing sphere about a cube of edge length ϕ {\displaystyle \phi }, and r i {\displaystyle r_{i}} is the apothem of a regular pentagon of edge length ϕ {\displaystyle \phi }.

The dihedral angle of a regular dodecahedron between every two adjacent pentagonal faces is 2 arctan (ϕ) {\displaystyle 2\arctan(\phi)}, approximately 116.565°.
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