In this paper a simulation of the properties of surface plasmons on gold nanoplatelets with various cross-sections inscribed in a circle and an investigation of their field distributions to assign multiple SPRs are described. The manipulated propagation can be obtained through the evolution of edges and corners. Furthermore, the particle morphology and the associated spectral positions alone do not uniquely reflect the important details of the local field distribution or the resonance modes. The plasmon modes were investigated and found to be mainly excited along the edges and in the side and sloped side surfaces. The strong field distributions can generally be found around the corners and how the plasmons transmit through the corners to adjacent edges was also investigated. Besides the plasmons excited along the edges as were found for the triangular nanoplatelets, plasmons were excited in the interior region of the triangular surfaces and were also investigated. Despite this in the infrared region, plasmon modes were found to be along the edges for the hexagonal nanoplatelets. Also, it can be seen that the change of nanoplatelet thickness can support different plasmon modes ranging from dipolar resonance mode to quadrupole resonance mode. The thickness far below the skin depth can display complex plasmon modes along the edges and on the side and sloping side surfaces as well as the strong coupling between the top and bottom surfaces. The observed plasmon resonance modes in this simulation reflect the interference of all these contributions including the plasmons along the edges and on the side surfaces. This is an essential step towards a thorough understanding of plasmon modes and the effect of edge and corner evolution in polygonous nanoplatelets.