The scattering features of a spheroidal particle illuminated by the Laguerre- Gaussian (LG) beam have been studied based on the generalized Lorenz-Mie theory. By using the localized approximations method, the beam shape coefficients are evaluated and the results obtained agree with the cases of on-axis incidence. Calculations of the far-field scattering intensity are performed to study the LG beam scattered by spheroids, of different size parameters and eccentricities. The simulation results show that when the particle's size parameter is within the range that can be compared to the wavelength of the incident light, the magnitude of the scattering intensity will increase as the particle's size parameter increases, and it will decrease as the ratio of the spheroid's major axis to minor axis increases. Comparisons between LG beams with different topological charges illumination are made and explained physically. It turns out that the magnitude of the scattering intensity decreases as the topological charge increases. The theoretical investigation in this paper may provide a more accurate particle model and reference for applications of LG beams in areas such as particle size measurement, atmospheric laser communication, atmospheric remote sensing and so on.