The spreading area of cells has been shown to play a central role in the determination of cell fate and tissue morphogenesis; however, a clear understanding of how spread cell area is determined is still lacking. The observation that cell area and force generally increase with substrate rigidity suggests that cell area is dictated mechanically, by means of a force-balance between the cell and the substrate. A simple mechanical model, corroborated by experimental measurements of cell area and force is presented to analyze the temporal force balance between the cell and the substrate during spreading. The cell is modeled as a thin elastic disc that is actively pulled by lamellipodia protrusions at the cell front. The essential molecular mechanisms of the motor activity at the cell front, including, actin polymerization, adhesion kinetics, and the actin retrograde flow, are accounted for and used to predict the dynamics of cell spreading on elastic substrates; simple, closed-form expressions for the evolution of cell size and force are derived. Time-resolved, traction force microscopy, combined with measurements of cell area are performed to investigate the simultaneous variations of cell size and force. We find that cell area and force increase simultaneously during spreading but the force develops with an apparent delay relative to the increase in cell area. We demonstrate that this may reflect the strain-stiffening property of the cytoskeleton. We further demonstrate that the radial cell force is a concave function of spreading speed and that this may reflect the strengthening of cell-substrate adhesions during spreading.