A meander is a (homotopy class of) a pair of curves on the sphere with transverse intersections. The meandric number M_n is the count of "rooted" meanders with 2n intersections. One has M_1=1, M_2=2, M_3=8 and M_4=42. Understanding the asymptotic growth rate of M_n remains an open problem in mathematics/statistical physics. In this talk we will consider the refinement M_{n,k} that counts meanders with 2n intersections and whose complement has exactly k bigons. In this settings, we can describe the asymptotics as n tends to infinity for each fixed k. Our solution relies on associating a quadratic differential to a meander (a so called square tiled surface) and relates the counting of M_{n,k} to the Masur-Veech volumes of quadratic differentials on the sphere with k punctures. This is a joint work with E. Goujard, P. Zograf and A. Zorich.