Solid angle orientation formula_9, and distance from the viewer as: where the surface area of a sphere is . The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2, is the area of a spherical cap on a unit sphere For small such that , this reduces to the area of a circle . The above is found by computing the following double integral using the unit surface element in spherical coordinates: This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the
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