Indentation of giant liposomes with a conical indenter is described by means of a tension-based membrane model. We found that nonlinear membrane theory neglecting the impact of bending sufficiently describes the mechanical response of liposomes to indentation as measured by atomic force microscopy. Giant vesicles are gently adsorbed on glassy surfaces via avidin-biotin linkages and indented centrally using an atomic force microscope equipped with conventional sharp tips mounted on top of an inverted microscope. Force indentation curves display a nonlinear response that allows to extract pre-stress of the bilayer T0 and the area compressibility modulus KA by computing the contour of the vesicle at a given force. The values for KA of fluid membranes correspond well to what is known from micropipet suction experiments and inferred from membrane undulation monitoring. Assembly of actin shells inside the liposome considerably stiffens the vesicles resulting in significantly larger area compressibility modules. The analysis can be easily extended to different indenter geometries with rotational symmetry.