2-dimensional (2D) nanosheets such as graphene, graphene oxide, boron nitride or transition metal dichalcogenides can be produced on a large scale by exfoliation techniques. The lateral shape of these 2D materials is typically considered random and irregular, and their average size is often estimated using techniques characterized by strong approximations or poor statistical significance. Here we measure in a quantitative, objective way the size and shape of 2D monoatomic nanosheets using a combination of optical, electronic and scanning probe techniques. We measure, one by one, the size and shape of thousands of sheets of graphene oxide as they undergo a standard ultrasonication treatment. Using automatic image processing and statistical modelling we identify two different fragmentation processes in 2D at the nanoscale, related to two populations of nanosheets described by gamma and exponential size distributions respectively. The two populations of sheets coexist during the fragmentation process, each one retaining its average size and shape. Our results explain the size reduction commonly observed in nanosheets upon sonication as an effect of changes in the respective weights of the two populations of nanosheets present in the material.