A polygonal body of revolution consisting of discrete meridional members joined by regular polygons is optimized for maximum drag and drag coefficient. The meridional, or primary members, carryall of the radial loads and the membrane structure transfers applied pressure loads to the primary members. The shape of the secondary membrane is found such that the performance index is optimized. Several methods are utilized to solve for the optimum conditions, and an application of these solutions to a parachute design is shown. The results are presented in parametric form with pressure distributions, structural flexibility, and skirt closure angle being the parameters investigated. In general, the results verified experimental data in several areas and that the current shapes used in design cover the optimum region, although refinements could be made in some areas. Of particular significance is the difference in shape for optimum drag and drag coefficient. For maximum drag force, the fullness or the lobing between meridional members should be a maximum throughout the span of the gore, whereas, for the maximum drag coefficient design, the lobing is a minimum in the region near the vent and gradually increases as the meridional distance increases. It was found that the pressure distribution influences the optimum design. For a typical distribution that increases as the meridional distance increases, the gore width should be increased near the skirt for an optimum drag coefficient. Parameter studies show that by increasing the secondary structure flexibility the drag increases, although it is a secondary effect for present designs. A well-known parameter in optimum parachute design is the suspension line length, and this study verifies that increasing the length improves the drag. Optimum conditions are found by use of the maximum principle, steepest descent, and the Rayleigh-Ritz method. / Ph. D.