Abstract The eigenvalue problem is considered for the Laplacian on regular polygons, with either Dirichlet or Neumann boundary conditions, which will be related to the unit circle by a conformal mapping. The polygonal problem is then equivalent to a weighted eigenvalue problem on the circle with the same boundary conditions. Upper bounds are found for the eigenvalues by the Rayleigh–Ritz method, where the trial functions are the eigenfunctions of the unweighted problem on the circle. These are products of Bessel and trigonometrical functions, and so the required integrals simplify greatly, with a new recursion formula used to generate some Bessel function integrals. Numerical results are given for the case of the hexagon with Dirichlet conditions. Consideration of symmetry classes makes computations more efficient, and gives as a byproduct the eigenvalues of a number of polygons, such as trapezoids and diamonds, which result from disecting the hexagon. Comparisons of the hexagon results are made with previous work.