  Cutting a hyperbolic surface X along a simple closed multi-geodesic results
in a hyperbolic structure on the complementary subsurface. We study the
distribution of the shapes of these subsurfaces in moduli space as boundary
lengths go to infinity, showing that they equidistribute to the Kontsevich
measure on a corresponding moduli space of metric ribbon graphs. In particular,
random subsurfaces look like random ribbon graphs, a law which does not depend
on the initial choice of X. This result strengthens Mirzakhani's famous simple
closed multi-geodesic counting theorems for hyperbolic surfaces.
