We all know if one draws a square, the √2 is an inherent value of the diagonal within the square, similarly if one draws a circle then л is the inherent value within the circle relating the diameter to circumference. Is foreknowledge of either of these factors is necessary for the completion of either task?
Given there is no direct evidence the Ancient Egyptians were cognizant of pi to say they were is an assumption.
Is it possible the ancient method given the method circumnavigate pi all together? Because we perceive pi is not evidence the Ancient Egyptians used pi in their calculations? What could explain this misconception is an assumption, on our part, that our concepts, principles and elements are compatible with the methods employed by the Ancient Egyptians?
According to the thought process demonstrated in Rhind Mathematical Papyrus, problem #50 the answer would be: No, they did not employ pi as we do calculating the area of a circle. Now I could be wrong, but I don't believe there is a single problem involving circles and circular geometric calculation noted in the papyri where they did not use the (8/9 d)^2 formula from problem #50 Rhind Mathematical Papyrus.
Based on this information one could logically conclude they had no knowledge of pi, or if they did, chose not to use it. This fact should be evident to all the only logical conclusion is they did not use pi!
If todays concepts principles and elements are applied to Ancient Egyptian methods we could then say they were cognizant of the following:
(8/9×2r)^2 = (256 r^2)/81 and creates a Geometric parabola
Derivative: d/dr(((8×2 r)/9)^2) = (512 r)/81
Integral: (256 r^2)/81 dr = (256 r^3)/243 + constant
lim_(r->∞) (256 r^2)/81 = ∞
After all there is just as much Ancient Egyptian evidence to support these equations as there is to support the use of pi or phi.
Truth be known, in spite of all available evidence or lack there of, the question still remains unanswered: Did the Egyptians really know pi? After all the value of 256/81 is a value we have assigned and really based on an extrapolated figure from a single process in the Rhind Mathematical Papyrus and is not fact but actually an assumption on our part.