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In the circular hard-disc cage approximation, moving the central disc towards a neighbour brings contact after a centre displacement $R-2r$. ThusAt close packing, $R=2r$ 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.

Insert the stipulated near-close-packing relation for $R$ into the hard-disc cage approximation:Writing $y=\sqrt{{\varphi }_{\mathrm{cp}}a/(N{a}_d)}$, we have ${\partial }_{a}y=y/(2a)$ and therefore ${\partial }_a\ln \mathcal{A}=y/[a(y-1)]$. From the supplied crystalline canonical partition function,Its close-packing scale is set by the triangular lattice packing of discs, with ${\varphi }_{\mathrm{cp}}=\pi /(2\sqrt{3})$ in the ideal geometry.