Using energy-filtered transmission electron microscopy we measured surface-plasmon resonances of gold nanoplatelets with different shapes and edge lengths at high spatial resolution. We find equidistant maxima of the energy-loss probability along the platelet edges. The plasmon dispersion of the different geometries is very similar, i.e., hardly dependent on specimen shape. The experimental results are verified by means of finite-difference time-domain calculations which reveal the presence of wedge-plasmon polaritons propagating along the platelet edges. At platelet corners, apart from radiative losses, wedge-plasmon polaritons are partially reflected or transmitted to neighboring edges. The interference of all these contributions leads to the observed plasmon resonance modes. This is an essential step towards a thorough understanding of plasmon eigenmodes in prismatic nanoplatelets.