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Yes. But the two ratios are abstract concepts, not physical objects. [Is there proof that pain exists?]. There are many mathematical series which, in the limit, equal pi. Two examples which are easy to remember, are:Gregory and Liebniz : 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - .... ), and
Euler: sqrt[6*(1^-2 + 2^-2 + 3^-2 + 4^-2 ... )]
I believe there is, yes, though I am not entirely sure what you mean with "existence". Pi can be defined as the ratio between the circumference of a circle, and its diameter. Of course, for a specific circle, this ratio exists. Well, this assumes that both the circumference and the diameter are well-defined. The diameter is pretty straightforward. For the circumference, you can use calculus to divide it into shorter and shorter pieces, take the limit, and prove that such a limit actually exists.You would then also need to prove that in Euclidian geometry, this ratio (circumference divided by diameter) is the same for any circle. I believe that wouldn't be too hard to prove. In fact, it's a general fact that similar figures have the same length ratios. For example, the ratio of diagonal to side-length is the same for all squares.
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