Solid angle surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as: Hence for a unit sphere the solid angle of the spherical cap is given as: When = , the spherical cap becomes a hemisphere having a solid angle 2. The solid angle of the complement of the cone (picture a melon with the cone cut out) is clearly: This is also the solid
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