We solve the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. By rescaling the temperature proportionally to the computational cell size ∆x (T → T ∆x/a_{eff} , where a eff is the lattice constant) [M. B. Hahn, J. Phys. Comm., 3:075009, 2019], we obtain Curie temperatures T_C that are in line with the experimental values for cobalt, iron and nickel. For finite-sized objects such as nanowires (1D) and nanolayers (2D), the Curie temperature varies with the smallest size d of the system. We show that the difference between the computed finite-size T_C and the bulk T_C follows a power-law of the type: (ξ_0 /d)^{λ} , where ξ_0 is the correlation length at zero temperature, and λ is a critical exponent. We obtain values of ξ_0 in the nanometer range, also in accordance with other simulations and experiments. The computed critical exponent is close to λ = 2 for all considered materials and geometries. This is the expected result for a mean-field approach, but slightly larger than the values observed experimentally.