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Regulate the surface at $z=ϵ$, so $r_{\max }=\sqrt{R^2-{ϵ}^2}$. On the hemisphere, $\sqrt{1+z^{\prime 2}}=R/z$, and its area isThe Ryu–Takayanagi formula therefore givesThe first term is the ultraviolet area-law divergence proportional to the boundary circle's length; the finite constant is universal for the vacuum of a three-dimensional conformal field theory.

Choose a constant-time boundary interval whose endpoints differ by $\pi$ in $\varphi$. Its Ryu–Takayanagi formula surface is the diameter through $r=0$. Cutting it off at $r_c=\pi /2-ϵ$, its length isThe holographic entanglement entropy is thereforewhere the Brown--Henneaux central charge $c=3R_{\mathrm{AdS}}/(2G)$ was used.

The Ryu–Takayanagi formula gives the leading entropy of a static boundary region:On the $t=0$ slice of Poincaré ${\mathrm{AdS}}_4$,The minimal surface anchored on a boundary circle of radius $R_1$ is the hemisphereCutting it off at $z=ϵ$, its area isThereforeThe universal requirement was the perimeter-law divergence $\pi R_1/(2Gϵ)$; the finite constant depends on the stated pure ${\mathrm{AdS}}_4$ vacuum geometry and equals $-\pi /(2G)$ here. Since $G\sim N^{-3/2}$, both terms are of order $N^{3/2}$.