A novel way to extract the Luttinger liquid parameter is proposed from a finite-size correction to the geometric entanglement per lattice site.This is achieved by exploiting a newly-developed scheme to efficiently compute the geometric entanglement per lattice site in the context of a matrix product state algorithm for finite systems with the periodic boundary conditions.The method is tested for a spin-1 XXZ model with ion anisotropy, which undergoes the Kosterlitz-Thouless transition for properly chosen parameters.We have succeeded in extracting the Luttinger liquid parameter in the critical regime, which is in a quantitative agreement with conformal field theory predictions, obtained through combining with the density matrix renormalization group.