= Solution
In the circular <hard-disc cage approximation>, moving the central disc towards a neighbour brings contact after a centre displacement $R-2r$. Thus
$$
\boxed{\mathcal A=\pi(R-2r)^2}.
$$
At close packing, \b[$R=2r$ and $\mathcal A=0$]. The radius concerns the accessible positions of the moving centre; subtracting only one disc radius would incorrectly leave a finite rattling area at contact. The real cage is bounded by curved exclusion arcs, so the circle is an inscribed approximation.
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