Consider the Poincaré upper half plane $\mathbb{H} = \{z \in \mathbb{C} \ : \ \Im(z) > 0\}$. This can be endowed with the Riemanian hyperbolic metric, defined by $$g_z(u,v) = \frac{1}{\Im(z)^2}\langle u, v \rangle$$ where $\langle \cdot, \cdot\rangle$ is the usual euclidian inner product, $z$ is a point of $\mathbb{H}$ and $u, v$ are tangent vectors at $z$, i.e. elements of $T_z \mathbb{H}$.
This Riemanian metric allows for instance to define a notion of length, integrating along a path the "local" norm of the speed. However, what about areas? I see that the formula for the hyperbolic area of a domain $B$ is often given to be $$\operatorname{Area}(B) = \int_B \frac{\mathrm dx \mathrm dy}{y^2}$$ but can this be deduced from the Riemannian data, i.e. the definition of $g_z$?