Gradient-based optimization of the aerodynamic performance, static stability, and stagnation-point heat transfer has been completed to find optimal heat shield geometries for blunt-body planetary entry vehicles. In the parametric study, performance trends have been identified by varying geometric parameters that define a range of cross-sections and axial shapes. Cross-sections considered include oblate and prolate ellipses, rounded-edge polygons, and rounded-edge concave polygons. Axial shapes consist of the spherical-segment, spherically-blunted cone, and power law. By varying angle-of-attack and geometric parameters, the aerodynamics, static stability, and heat transfer are optimized based on Newtonian impact theory with semi-empirical shock-standoff distance and stagnation-point heat transfer correlations. Methods have been verified against wind tunnel and flight data of the Apollo Command Module and are within 15% for aerodynamic coefficients and stagnation-point heat fluxes. Results indicate that oblate parallelogram configurations provide optimal sets of aerothermodynamic characteristics.