Platonic solid the inradius "r". That is, 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. The constants "φ" and "ξ" in the above are given by 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
Surface energy radius formula_1 and length formula_2 at high temperature and a constant uniaxial tension formula_3, then at equilibrium, the variation of the total Helmholtz free energy vanishes and we have where formula_5 is the Helmholtz free energy and formula_6 is the surface area of the rod: Also, since the volume (formula_8) of the rod remains constant, the variation (formula_9) of the volume is zero, i.e., Therefore, the surface energy density can be expressed as The surface energy density of the solid can be computed by measuring formula_3, formula_1, and formula_2 at equilibrium. This method is valid only if the solid is