We report on structuring-induced changes of the magnetic anisotropy of cylindrical nanostructures which are carved out of thin Pt/Co/Pt films. The magnetic properties of films and structures with a diameter of about 34 nm were investigated via magneto-optic Kerr effect. The magnetic anisotropy is determined for both films and nanostructures for varying Co thicknesses (0.5–7 nm). In general, the nanostructures exhibit larger perpendicular anisotropy than the films. On thickness increase of the Co layer two spin reorientation transitions at about 2.2 and 5 nm are found. At 2.2 nm the nanostructures exhibit the transition from perpendicular to in-plane orientation of magnetization while at 5 nm the reversed transition is found. The variation of the magnetic anisotropy of the Co nanostructures is not solely caused by the change of shape anisotropy. The net change, corrected for the shape, reveals a reduction of strain in the thinnest Co layers while the increase of the anisotropy of the nanostructures at higher Co thicknesses is caused by a transformation of the Co lattice from fcc to hcp.