The magnetic anisotropy of Co/Pt multilayers is investigated. The perpendicular uniaxial anisotropy is discussed in second order approximation under a variation in Co and Pt layer thicknesses. The evolution of anisotropy constants is shown in the phase diagram of first and second order anisotropy constants. A thickness driven spin reorientation via the canted phase is observed for a single Co layer as well as for Co/Pt multilayer films. (C) 2009 American Institute of Physics. {[}DOI: 10.1063/1.3070644]