El dodecaedre regular està format per dotze pentàgons regulars.

Característiques bàsiques

  • Cares = 12
  • Arestes = 30
  • Vèrtexs = 20

Coordenades dels vèrtexs

Per un dodecaedre regular d'aresta unitat, les coordenades dels seus vèrtexs són:

  • \(V_1=\left(0,a,c\right)\)
  • \(V_2=\left(0,a,-c\right)\)
  • \(V_3=\left(0,-a,c\right)\)
  • \(V_4=\left(0,-a,-c\right)\)
  • \(V_5=\left(c,0,a\right)\)
  • \(V_6=\left(c,0,-a\right)\)
  • \(V_7=\left(-c,0,a\right)\)
  • \(V_8=\left(-c,0,-a\right)\)
  • \(V_9=\left(a,c,0\right)\)
  • \(V_{10}=\left(a,-c,0\right)\)
  • \(V_{11}=\left(-a,c,0\right)\)
  • \(V_{12}=\left(-a,-c,0\right)\)
  • \(V_{13}=\left(b,b,b\right)\)
  • \(V_{14}=\left(b,b,-b\right)\)
  • \(V_{15}=\left(b,-b,b\right)\)
  • \(V_{16}=\left(b,-b,-b\right)\)
  • \(V_{17}=\left(-b,b,b\right)\)
  • \(V_{18}=\left(-b,b,-b\right)\)
  • \(V_{19}=\left(-b,-b,b\right)\)
  • \(V_{20}=\left(-b,-b,-b\right)\)

on:

  • \(a=\frac{1}{2}\)
  • \(b=\frac{1+\sqrt{5}}{4}\)
  • \(c=\frac{3+\sqrt{5}}{4}\)

Característiques avançades

Per un dodecaedre regular d'aresta unitat tenim:

  • Rcircumscrita = \(\frac{\sqrt{3}+\sqrt{15}}{4}\)
  • Rtangent arestes = \(\frac{3+\sqrt{5}}{4}\)
  • Rinscrita \(\frac{\sqrt{10\left(25+11\sqrt{5}\right)}}{20}\)
  • Volum = \(\frac{15+7\sqrt{5}}{4}\)

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