  Let $p$ be a branched covering of a Riemann surface to the Riemann sphere
$\mathbb{P}^1$, with branching set $B \subset \mathbb{P}^1$. We define the
complexity of $p$ as infinity, if $\mathbb{P}^1 \setminus B$ does not admit a
hyperbolic structure, or the product of its degree and the hyperbolic area of
$\mathbb{P}^1 \setminus B$, otherwise. The topological complexity of a surface
$S$ is defined as the infimum of the set of all complexities of branched
coverings $M \to \mathbb{P}^1$, where $M$ is a Riemann surface homeomorphic to
$S$. We prove that if $S$ is a connected, closed, orientable surface of genus
$g$, then its topological complexity, $C_{\text{top}}(S)$, is given by:
  \[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2\pi(2g+1) & \mbox{if } g \geq
1, 6 \pi & \mbox{if } g=0. \end{array} \right.\]
