We present a theoretical study of the shape and free energy of a vesicle (or microemulsion droplet) adhered to a substrate (other droplet) based on the expression for the surface free energy by Helfrich. Analytical formulas are presented for the shape and free energy when the rigidity constant for bending, k, is small; i.e., when (k/sigma)(1/2), with sigma the surface tension, is small compared to the typical dimension of the vesicle (k/sigma)(1/2) much less than V-1/3, with V the vesicle volume. These formulas are compared with numerical solutions of the shape equations such as those first provided in the work by Seifert and Lipowsky. Results are presented when the exact formulas are applied to study the onset of microemulsion droplet aggregation, e.g., dimer formation, in terms of the usual coefficients in the Helfrich free energy expression, such as the rigidity constant for bending and the spontaneous curvature.