  We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold
is equal, up to an additive constant, to the volume of its convex core. We also
provide a precise upper bound on the renormalized volume in terms of the
Weil-Petersson distance between the conformal structures at infinity. As a
consequence we show that holomorphic disks in Teichm\"uller space which are
large enough must have "enough" negative curvature.
