Groemer Theorem

Given $n$ Circles and a Perimeter $p$, the total Area of the Convex Hull is

$$A_{\rm Convex\ Hull} = 2\sqrt{3}(n-1)+p(1-{\textstyle{1\over 2}}\sqrt{3}\,)+\pi(\sqrt{3}-1).$$

Furthermore, the actual Area equals this value Iff the packing is a Groemer Packing. The theorem was proved in 1960 by Helmut Groemer.

See also Convex Hull