We have studied magnetic anisotropies of Fe atoms on the platinum (111) surface, employing a fully relativistic pseudopotential and plane wave method with the local spin density approximation. We investigated three surface structures with different Fe monolayer coverages: full coverage, half-coverage and quarter-coverage. The effect of surface relaxation has been included. It was found that the magnetic easy axis of the system is within the surface plane for all systems investigated. In the system of an Fe chain on Pt(111), having an anisotropic local structure, the magnetic anisotropy energy is much enhanced after surface relaxation. This absolute value is larger compared with the value for bulk alloy and the magnetic easy axis is directed parallel to the alignment of Fe atoms.