The classical theory of small amplitude shallow water waves is applied to regular polygonal basins. The natural frequencies of the basins are related to the eigenvalues of the Helmholtz equation. Exact solutions are presented for triangular, square and circular basins while pentagonal, hexagonal and octagonal basins are solved, for the first time, by an efficient Ritz method. The first five eigenvalues are tabulated and the corresponding mode shapes are discussed. Tileability conditions are presented. Some modes can be tiled into larger domains.