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Numerical results for elliptical waveguides are usually obtained from the appropriate Mathieu and associated Mathieu functions, but these are expensive to compute, because of slow convergence, and difficult to tabulate. It is shown that results for the cutoff wavelengths can be obtained using simple new formulas relating the ellipse (or circle) to an inscribed polygon. Cutoff wavelengths for the polygon can then be obtained efficiently using an available computer program. In this way, by applying boundary perturbation theory, the accuracy for cutoff wavelength from the original program has been increased by a factor of up to 104. Besides using an extrapolation technique, special advantage is taken of the elliptical shape which allows an `area correctionÂ¿ to be applied to the approximating polygonal results.