In this paper, the contribution of magnetoelastic-anisotropy effect on perpendicular-magnetic anisotropy of Co/Pt superlattice thin films is studied considering the strain-dependent elastic constants. We measured the effective-magnetic-anisotropy energy changing the Pt-layer thickness (2 to 20 Å) keeping the Co-layer thickness unchanged (4 Å). The effective-magnetic-anisotropy energy changed depending on the Pt-layer thickness and showed a maximum near dPt=12 Å. This result is explained by calculating the magnetoelastic-anisotropy energy considering the change of the elastic constants caused by the lattice misfit. We measured anisotropic elastic constants of Co/Pt superlattice thin films by resonance-ultrasound-spectroscopy method coupled with laser-Doppler interferometry and the picosecond laser-ultrasound method. The measured elastic constants showed elastic anisotropy between the in-plane and out-of-plane directions. They agreed with those predicted from the strain-dependent elastic-constant model. The elastic constants show correlations with the perpendicular-magnetic anisotropy.