Is there an alternative to using pi in calculating area/circumference?

 

I'm curious if a method has ever been explored that can measure the area of a circle as perfectly as a square, even if it is complicated.

Basicaly you are restating the problem of squaring the circle and the impossibility of this undertaking follows from the fact that π (pi) is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. The transcendence of π was proved by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible.

Seeing as the definition of pi is that the circumference of a circle of radius r is 2*pi*r, then obviously the answer is no. Of course in practical considerations we always use some rational approximation (such as 3.14) so the answer is yes. Or we could conclude that the question is ambiguous.

For the sake of arguement, there are many physical ways to do these things. But they are not mathematical really.