This report deals with a method of evaluating coupled natural frequencies of a circular cylindrical tank. Starting from a variational principle, the authors obtain a functional relating to coupled oscillations between a linear elastic body of infinitesimal deformations and a perfect fluid of small wave heights. By imposing as constraints several of stationary conditions of the functional above-mentioned, an alternate functional is derived from which a complementary Rayleigh quotient is obtained. The first approximation for the complementary Rayleigh quotient gives approximate formulas of coupled natural frequencies for a circular cylindrical tank oscillating in an axisymmetric manner. Numerical examples show satisfactory coincidence between exact natural frequencies and approximate ones in the case of sloshing modes, whereas tolerable error of the approximation in the case of lower bulging modes.