  We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the
moduli space of quadratic differentials of unit area on curves of genus $g$
with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms
of a family of polynomials in $n$ variables governed by the topological
recursion/Virasoro constraints. This is equivalent to a formula giving these
polynomials as a sum over stable graphs, and retrieves a result of
\cite{Delecroix} proved by combinatorial arguments. Our method is different: it
relies on the geometric recursion and its application to statistics of
hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain
an expression of the area Siegel--Veech constants in terms of hyperbolic
geometry. The topological recursion allows numerical computations of
Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and
$n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We
also relate our polynomials to the asymptotic counting of square-tiled surfaces
with large boundaries.
