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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

© 1996-9 Eric W. Weisstein