A vast range of both living and inanimate planar cellular partitions obeys universal empirical laws describing their structure. To better understand this observation, we analyze the morphometric parameters of a sizeable set of experimental data that includes animal and plant tissues, patterns in desiccated starch slurry, suprafroth in type-I superconductors, soap froths, and geological formations. We characterize the tilings by the distributions of polygon reduced area, a scale-free measure of the roundedness of polygons. These distributions are fairly sharp and seem to belong to the same family. We show that the experimental tilings can be mapped onto the model tilings of equal-area, equal-perimeter polygons obtained by numerical simulations. This suggests that the random two-dimensional patterns can be parametrized by their median reduced area alone.