Forget equations for a moment...as surely we can add enough fractions together and apart to define pi probably down to its millionth part, and the AEs knew what 'millions' were.
What about scale?
Anyone can make a perfect circle using a pivot-point and something to extend from that point, and just turn about the pivot-point.
What if I double the extension? Triple it? Go out four hundred and 19 feet and the width of my eyelash?
What if I get several perfectly formed, square-edged sticks all the same length, and place them on the edges of the circle so that the Sun passes right over them and they cast no shadows upon the ground...and I measure the time it takes for no shadow to appear on one stick of a circle and then not appear on the other stick. And I do this to the four circles I have made.
I know the diameters of the circles, and now I can equate their sizes to how much time passes in shadow-creations because I have made the circles all from the same pivot-point...and strung a simple line across all the upright sticks...and can measure the distances between the sticks and show that 'time' equals the Sun moving 'x' distances according to my circles and their sticks and shadows not being cast.
Have another think about a process for laying out the foundations of temples and such, where the Pharaoh or High Priest uses a string line to measure against the stars...or a star...or a certain source of light.
Maybe its time we rethink pi.
As always, Cheers