Since the development of the generalized Lorenz-Mie theory, electromagnetic scattering by arbitrary beams has drawn growing interest. The Laguerre-Gaussian (LG) vortex beam is well known for its orbital angular momentum. With the aim of investigating the analytical solution to the scattering of a chiral sphere by a LG vortex beam, particular attention is paid to the expansion expression of the LG vortex beam. The expansion coefficients are derived based on the expansion of a Hermite Guassian beam as the LG vortex beam can be expressed as the superposition of Hermite Guassian modes. The numerical results of the incident beam expansion coefficients convergence and the scattered field comparison with the reference prove the validity of the theoretical analysis and computation codes. The results reveal that there exists an optimal sphere size for the maximum scattered field which is determined by the topological charge, beam waist radius, and beam center position. The investigation could provide a foundation for the optical manipulation of chiral particles by a LG vortex beam.