Perfect optical vortex (POV) beams have attracted extensive attention because they have the advantage of a radial profile that is independent of orbital angular momentum. To date, it is usually obtained by means of the Fourier transform performed by a lens on Bessel beams. We theoretically and experimentally demonstrate that POV can be generated by performing the Fourier transform on Laguerre-Gauss beams with a high-order radial index. Furthermore, we derive an analytical expression for the increase in vortex radius, which is beneficial to compensate for the influence of the radius change in actual experiments. Our results may shed new light for a variety of research utilizing POV.