The shapes of several urchins are correctly predicted by a model that uses only measured height and diameter as fitted variables. The predicted shape is based on the engineering theory of thin shells, which is conventionally used to calculate the shapes of a fluid droplet on a horizontal plane, and of `buckle-free' engineered domes. The magnitudes of forces theoretically required to generate urchin shapes at realistic sizes are similar to forces typically exerted on the skeleton by self-weight, podia and coelomic pressure. An urchin's shape, despite complex details of plate growth, is thus determined by a force balance at each point in the skeleton. Despite the skeleton's apparent rigidity, over developmental time it must deform in a manner similar to a stretchy balloon. This membrane model of morphogenesis specifies two developmental shape parameters: (i) the apical curvature; and (ii) a ratio of the mimicked vertical gradient of pressure (podial forces, etc.) to the internal coelomic pressure. An ontogenetic series of urchins is represented as a curved line in a two-dimensional, developmental morphospace. This morphospace, which is useful for studying developmental constraints and macroevolutionary dynamics, explains observed patterns of allometry in height and diameter in urchins.