Cells are wrapped in inelastic membranes, yet they can sustain large mechanical strains by regulating their area. The area regulation in cells is achieved either by membrane folding or by membrane exo- and endocytosis. These processes involve complex morphological transformations of the cell membrane, i.e., invagination, vesicle fusion, and fission, whose precise mechanisms are still under debate. Here we provide mechanistic insights into the area regulation of cell membranes, based on the previously neglected role of membrane confinement, as well as on the strain-induced membrane tension. Commonly, the membranes of mammalian and plant cells are not isolated, but rather they are adhered to an extracellular matrix, the cytoskeleton, and to other cell membranes. Using a lipid bilayer, coupled to an elastic sheet, we are able to demonstrate that, upon straining, the confined membrane is able to regulate passively its area. In particular, by stretching the elastic support, the bilayer laterally expands without rupture by fusing adhered lipid vesicles; upon compression, lipid tubes grow out of the membrane plane, thus reducing its area. These transformations are reversible, as we show using cycles of expansion and compression, and closely reproduce membrane processes found in cells during area regulation. Moreover, we demonstrate a new mechanism for the formation of lipid tubes in cells, which is driven by the membrane lateral compression and may therefore explain the various membrane tubules observed in shrinking cells.