Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory(RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequency for a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.