The Area method for phase stability analysis is re-examined from the viewpoint of the topology of thermodynamic surfaces. A new generalization of the Area method objective function and the maximum area criterion is proposed in terms of the convex envelope of thermodynamic surfaces. Several special cases for two-component, single- and two-phase systems are analyzed and the results generalized to hypersurfaces, characteristic of the Gibbs energy of multicomponent, multiphase systems. It is shown conclusively that the area (or more generally hypervolume) between the free energy hypersurface and a hyperplane that intersects the hypersurface is maximal when the hyperplane coincides with the convex envelope of the hypersurface. Extension of the maximum area criterion to four basic thermodynamic surfaces for pure fluids and mixtures is discussed. The authors believe the approach adopted here sets the Area method on secure theoretical foundations, answers criticisms concerning its validity, and will stimulate fresh research.