Cells and organelles are shaped by the chemical and physical forces that bend cell membranes. The human red blood cell (RBC) is a model system for studying how such forces determine cell morphology. It is thought that RBCs, which are typically biconcave discoids, take the shape that minimizes their membrane-bending energies, subject to the constraints of fixed area and volume. However, recently it has been hypothesized that shear elasticity arising from the membrane-associated cytoskeleton (MS) is necessary to account for shapes of real RBCs, especially ones with highly curved features such as echinocytes. In this work we tested this hypothesis by following RBC shape changes using spherical harmonic series expansions of theoretical cell surfaces and those estimated from 3D confocal microscopy images of live cells. We found (i) quantitative agreement between shapes obtained from the theoretical model including the MS and real cells, (ii) that weakening the MS, by using urea (which denatures spectrin), leads to the theoretically predicted gradual decrease in spicule number of echinocytes, (iii) that the theory predicts that the MS is essential for stabilizing the discocyte morphology against changes in lipid composition, and that without it, the shape would default to the elliptocyte (a biconcave ellipsoid), (iv) that we were able to induce RBCs to adopt the predicted elliptocyte morphology by treating healthy discocytes with urea. The latter observation is consistent with the known connection between the blood disease hereditary elliptocytosis and spectrin mutations that weaken the cell cortex. We conclude that while the discocyte, in absence of shear, is indeed a minimum energy shape, its stabilization in healthy RBCs requires the MS, and that elliptocytosis can be explained based on purely mechanical considerations.