A propeller is more than just an "air pump" or a fan. A prop is shaped like a wing, and low pressure on top (front side of the prop) typically creates 2/3 to 3/4 of the pulling force. Only 1/4 to 1/3 is created by air being blown backwards. These numbers apply to full size airplanes like Cessna 172s and Piper Cherokees, but the principles are the same. The shape of the prop airfoil decides how the prop will perform. Thus calculating the volume of a cylinder won't be very helpful.

Back to the original poster's question: Comparing two props of different diameter, the larger-diameter prop will *generally* be more efficient; it will turn a higher percentage of the engine power into thrust. Some power is always "wasted" (noise/turbulence/...), and larger props tend to run "cleaner". Generally speaking. Pe Reivers' spreadsheet has been mentioned, and I recommend that you experiment with different prop sizes and see how the results change.

The 12x8 vs the 11x8: The 12x8 will most likely produce more thrust, and the 11x8 will most likely give you more speed. In theory. Type of airplane and engine will play a major part. The 12x8 will produce more thrust if the engine can pull it. My Irvine 53 will probably produce more thrust with the 11x8 because it lets it rev up to "where the power is".

If you go down in pitch to a 12x6, rpms will be closer to the 11x8. The 12x6 will definitely produce more thrust than the 11x8, but top speed will be lower.

Hope this helps more than it confuses.