We study the Masur-Veech volumes MVg,n$MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g$g$ with n$n$ punctures. We show that the volumes MVg,n$MV_{g,n}$ are the constant terms of a family of polynomials in n$n$ 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 [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] 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 [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. 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$g$ and n$n$, which leads us to propose conjectural formulae for low g$g$ but all n$n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.
