Flexible phased arrays potentially enable diverse applications not permitted by rigid systems; however, they introduce ambiguity in antenna element positions. If this position ambiguity can be overcome, flexible arrays can perform the full suite of array functions: beam steering, wavefront engineering, and beam focusing. Furthermore, shape reconstructions of arrays can be used for applications beyond beamforming. We propose a framework to reconstruct the shape of a flexible array that only uses mutual coupling measurements and does not require additional sensors or functionalities in the system. We discuss the approach, a two-step algorithm, which is highly modular and can be implemented in a variety of phased array systems. To demonstrate the accuracy of the approach, we present results from two passive 2.5-GHz phased array setups using dipole and patch antennas, as well as a 10-GHz (active) integrated circuit flexible phased array, and demonstrate the accuracy of the approach in this system. In all cases, the algorithm reconstructs the antenna shape accurately, with average position errors of approximately 6% of the wavelength. This article can serve as the beginning of the broad study of shape reconstruction algorithms and their applications.