  We prove that any length metric space homeomorphic to a 2-manifold with
boundary, also called a length surface, is the Gromov-Hausdorff limit of
polyhedral surfaces with controlled geometry. As an application, using the
classical uniformization theorem for Riemann surfaces and a limiting argument,
we establish a general "one-sided" quasiconformal uniformization theorem for
length surfaces with locally finite Hausdorff 2-measure. Our approach yields a
new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular
quasispheres.
