Generation of tissue curvature is essential to morphogenesis. However, how cells adapt to changing curvature is still unknown because tools to dynamically control curvature in vitro are lacking. Here we developed self-rolling substrates to study how flat epithelial cell monolayers adapt to a rapid, anisotropic change of curvature. We show that the primary response is an active and transient osmotic swelling of cells. This cell volume increase is not observed on inducible wrinkled substrates, where concave and convex regions alternate each other over short distances, identifying swelling as a collective response to changes of curvature with persistent sign over large distances. It is triggered by a drop in membrane tension and actin depolymerization, perceived by cells as a hypertonic shock. Osmotic swelling restores tension while actin reorganizes, probably to comply with curvature. Epithelia are thus unique materials that transiently, actively swell while adapting to large curvature induction.