Rayleigh and Rayleigh-Ritz represent the two approximate analytical methods which can be used in this paper in order to study the characteristics of free transverse vibrations of cantilever beam in the two cases. The first one is at tapered thickness and constant width while the second case is at tapered width and. The free end of the cantilever is sharp in both cases and is of different values of length. The study selects the value of thickness at clamped end (hc) and change the values of width at this end (wc) with thickness ratio which equals to ( 0.1 – 5 ). The second case on the other hand, can select the value of width (wc) at clamped end then change the thickness (hc) at ratio ( 0.1 – 5 ). Through out the results, it is shown that the cantilever beam at tapered thickness has natural frequency lesser than that of tapered width of the same length and at the same dimensions of clamped end, the natural frequency decrease with increasing the length of cantilever beam also with decreased the value of tapered thickness or tapered width according to this study. In case of different values of tapered width for the same length of beam and the same value of thickness at clamped end the cantilevers have the same value of frequency, also the cantilevers have constant frequency at tapered thickness for different values of ( ), while in the case of tapered width the frequency increases with increasing the value ( ) for the same (wc) and at the same length of cantilever beam.