The main result of this paper is an upper bound to the weighted density \(D\) of a packing of circles on the sphere, with radii selected from a given set \(\{r_{1},\ldots ,r_{n}\}\), namely \(D\leq \max \{D(r_{i},r_{j},r_{k})\), \(1\leq i\leq j\leq k\leq n\}\), where \(D(r_{i},r_{j},r_{k})\) denotes the weighted density of three mutually touching circles with radii \(r_{i},r_{j},r_{k}\), with respect to the triangle spanned by the centers of the circles. “Weighted density” means that a positive weight is assigned to each radius \(r_{i}\) which is taken into account in the calculation of the density. The upper bound is attained if and only if the packing is saturated and the associated Delaunay triangulation consists of extremal triangles, i.e. triangles with maximal weighted density. This theorem may be used, for instance, to prove that the system of incircles of the Archimedean tiling \((4,4,n)\) with \(n\geq 6\) has maximal ordinary density.
The concept of weighted density is also used to prove a sufficient condition for the solidity of a packing of circles on the sphere. (A packing of circles is called solid if no finite number of its members can be rearranged so as to form, together with the rest of the members, a packing not congruent to the original one.) As a consequence, it is proved that the incircles of the faces of the Archimedean tilings \((4,6,6)\), \((4,6,8)\), and \((4,6,10)\) form solid packings.