"Is it the added thickness that must be penetrated, or the increase in area the force of the round is applied to?"
By increasing the angle of the armor you increase its effeciency, thereby decreasing the amount of plating needed to do the job. I don't have the equations handy so these numbers are not accurate, just an illustration. Say, for intance you have 60mm of armor at 0 degrees, 60mm of plating angled at 45 degrees would give twice protection. You can substitute thickness by angling your armor. The inverse is also true you can decrease the thickness of the plating and still have the same protection. But back to your question...the thicker ( or more angled) the plating the more energy is required to penetrate. At the risk of contradicting myself and muddying the waters the thickness(and angling) is more important. You can have a very small round (37 or 20mm) fired at a very high velocity and it will not penetrate because it will lose its kinetic energy very quickly. If you shoot a large round( 75 or 88mm) round at the same velocity, it will have more kinetic enegry due to the increase in mass making penetration more likely.
Please forgive my verbosity and I hope this helps.