We measure the geometry of a crumpled sheet of paper with laser-aided topography and discuss its statistical properties. The curvature of an elastoplastic fold scales linearly with applied force. The curvature distribution follows an exponential form with regions of high curvature localized along ridges. The measured ridge length distribution is consistent with a hierarchical model for ridge breaking during crumpling. A large fraction of the ridges are observed to terminate without bifurcating, and the ridge network connectedness is not as complete as anticipated. The self-affinity of the surface is characterized by a Hurst exponent of 0.71+/-0.01 in contrast with previous results.