>I agree with that. The majority (if not all) of Euclid's elements seems to be just that. You may have just found your answer.
Why not just observe that all statements about area (given in unit square units) are in fact equivalent to statements about ratios to the unit square. That is, saying Area(X) = 57 means X has 57 times the area of the unit square. Then everything else "falls out" as we used to say (do they still?) Complicated reworkings of Riemann integrals are absolutely not needed.
