  We study the volume entropy of a class of presentations (including the
classical ones) for all surface groups, called \emph{minimal geometric
presentations}. We rediscover a formula first obtained by Cannon and Wagreich
with the computation in a non published manuscript by Cannon. The result is
surprising: an explicit polynomial of degree $n$, the rank of the group,
encodes the volume entropy of all classical presentations of surface groups.
The approach we use is completely different. It is based on a dynamical system
construction following an idea due to Bowen and Series and extended to all
geometric presentations in. The result is an explicit formula for the volume
entropy of minimal presentations for all surface groups, showing a polynomial
dependence in the rank $n>2$. We prove that for a surface group $G_n$ of rank
$n$ with a classical presentation $P_n$ the volume entropy is
$\log(\lambda_n)$, where $\lambda_n$ is the unique real root larger than one of
the polynomial \[ x^{n} - 2(n - 1) \sum_{j=1}^{n-1} x^{j} + 1. \]
