This paper presents a general method to perform simultaneous topological and dimensional synthesis for planar rigid-body morphing mechanisms. The synthesis is framed as a multi-objective optimization problem for which the first objective is to minimize the error in matching the desired shapes. The second objective is typically to minimize the actuating force/moment required to move the mechanism, but different applications may require a different choice. All the possible topologies are enumerated for morphing mechanism designs with a specified number of degrees of freedom (DOF), and infeasible topologies are removed from the search space. A multi-objective genetic algorithm (GA) is then used to simultaneously handle the discrete nature of the topological optimization and the continuous nature of the dimensional optimization. In this way, candidate solutions from any of the feasible topologies enumerated can be evaluated and compared. Ultimately, the method yields a sizable population of viable solutions, often of different topologies, so that the designer can manage engineering tradeoffs in selecting the best mechanism. Three examples illustrate the strengths of this method. The first examines the advantages gained by considering and optimizing across all the topologies simultaneously. The second and third demonstrate the method's versatility by incorporating prismatic joints into the morphing chain to allow for morphing between shapes that have significant changes in both shape and arc length.