Bulk and nanoparticle powders of LaCoO3 (LCO) were synthesized, and their magnetic and structural properties were studied using SQUID magnetometry and neutron diffraction. The bulk and large nanoparticles exhibit weak ferromagnetism (FM) below T ~ 85K and a crossover from strong to weak antiferromagnetic (AFM) correlations near a transition expressed in the lattice parameters, To ~ 40K. This crossover does not occur in the smallest nanoparticles; instead, the magnetic behavior is predominantly ferromagnetic. The amount of FM in the nanoparticles depends on the amount of Co3O4 impurity phase, which induces tensile strain on the LCO lattice. A core-interface model is introduced, with the core region exhibiting the AFM crossover and with FM in the interface region near surfaces and impurity phases.