We demonstrate that in nanometer-size magnets the superposition of the lattice dependence of the dipolar energy and the truncation of dipolar sums leads to size- and lattice-dependent effective perpendicular anisotropy. As a consequence, the spin reorientation transition in small platelets of identical shape on different lattices occurs at different sizes for identical anisotropy energy. In contrast to conventional results influences of size on the magnetic behavior can be found even at large aspect ratios of size to thickness.