We investigated the magnetic switching of MgO/Co/Pt pillars with perpendicular magnetic anisotropy, for lateral pillar sizes from 30 nm to 2 lm and for Co layer thicknesses between 1.8 and 2.6 nm. For large pillars, both the coercivity and the remanent magnetization decrease for increasing Co thickness. For all Co thicknesses, the coercivity strongly increases upon decreasing the pillar size. A comparison with micromagnetic simulations shows that the change in coercivity is determined by size-dependent demagnetizing effects. Our results show that small pillars with perpendicular magnetization and a tunable coercivity can be fabricated from continuous layers with in-plane magnetization.