Pi is hard to integrate currently because it moves off the page in 2D both in and out while you rotate with it. We don't see this when we draw a circle. We pretend that the exponential pi doesn't do this whereas a true representation of Pi would be a spiral in and then out of the page over 4 consecutive half turns.

This is the reason why Pi X R squared isn't Pi squared R squared which is what you would get with the constant of 7 or 10 or 200

Square, side 20cm, perimeter 80 cm, 400 cm^2

Square, side 200cm, perimeter 800 cm, 40000 cm^2

The square's magnitude gets squared and the UNITS change from cm to cm^2, same for volume to cm^3

Circle, side 20, perimeter 2Pi X 40, area Pi X 400cm^2

Circle, side 200, perimeter 2Pi X 400, area Pi X 40000cm^2

No unit change... Pi is an exponent of itself, not a constant in the normal way.

When you integrate dimensions or units they change from x to x^2, but not Pi. The only other unit that does this is our unit ---- 1 unit, if we used units of two, then the unit squared would have a square relationship. If we fixed Pi as the unit, 1 would become irrational with respect it.

That is why you use a unit. 2Pi is used as a unit everywhere as a constant. It seems to behave like one.

Yet it is Pi and not 4Pi^2 with the area of a circle compared to the circumference like with the other constants 20 and 200 they increase 10 fold and so did the unit, they got squared.

Pi is it own exponent in that if you multiply it by 2 it doubles. But when we double it (by making a circle with a radius twice that before), we differentiate Pi at the same time from 2Pi to 2pi/2 = pi.

That is why 2Pi does not behave like a normal constant.

And that is why Pi circles the square