Platonic solid to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most. Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs. If a polyhedron has Schläfli symbol {"p", "q"}, then its dual has the symbol {"q", "p"}. Indeed, every combinatorial property of one
Platonic solid by where "θ" is the dihedral angle. The midradius "ρ" is given by where "h" is the quantity used above in the definition of the dihedral angle ("h" = 4, 6, 6, 10, or 10). Note that the ratio of the circumradius to the inradius is symmetric in "p" and "q": The surface area, "A", of a Platonic solid {"p", "q"} is easily computed as area of a regular "p"-gon times the number of faces "F". This is: The volume is computed as "F" times the volume of the pyramid whose base is a regular "p"-gon and whose height is