= Solution
Regulate the surface at $z=\epsilon$, so $r_{\max}=\sqrt{R^2-\epsilon^2}$. On the hemisphere, $\sqrt{1+z'^2}=R/z$, and its area is
$$
\begin{aligned}
\operatorname{Area}(\gamma_D)
&=2\pi R\int_0^{r_{\max}}
\frac{r\,dr}{(R^2-r^2)^{3/2}}\\
&=2\pi\left(\frac R\epsilon-1\right).
\end{aligned}
$$
The <Ryu–Takayanagi formula> therefore gives
$$
\boxed{S_D=\frac{\pi}{2G}
\left(\frac R\epsilon-1\right)}.
$$
The 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.
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