theory through a computation in the bulk. In particular $$\label{RT prescription}
S_A = \frac{\textrm{Area}(\gamma_A)}{4G_N^{(d+2)}}$$ where $G_N^{(d+2)}$ is the Newton constant in $d+2$ spacetime dimensions.\
Depending on the shape of $\partial A$, one decides if it is more convenient to work either in cartesian ($d\vec{x}^2=\sum_{i=1}^d (dx^i)^2$) or polar coordinates ($d \vec{x}^2=d\rho^2+\rho^2 d\Omega^2_{d-1}$, being $d\Omega^2_{d-1}$ the metric of the $d-1$ dimensional unit sphere) of $\mathbb{R}^d$. Since we will mostly consider $A$ to be a finite strip or a disjoint union of two of them, we will adopt the cartesian coordinates for $\mathbb{R}^d$ (for an example where the polar coordinates system is employed, see the appendix \[app circle\], which contains a discussion on the circular case in the black hole background considered below).\
For many known black holes which are asymptotically $AdS_{d+2}$, the metric on the fixed time slice is given by $$\label{ds20 ansatz}
ds^2_0 \equiv
ds^2 \big|_{t = \textrm{const}} 
= R^2\left(
\frac{d\vec{x}^2}{z^2}+\frac{dz^2}{z^2f(z)}\right)$$ where $R$ is the radius of $AdS_{d+2}$ realized closed to the boundary. In this system of coordinates the boundary is the $z=0$ slice and the horizon is characterized by the smallest zero of the emblacking function $f(z)$.\
Let us consider the region $A$ in the boundary given by a strip 