>>1
By "infinite" one can only hope you only failed to say that it has an infinite number of decimal places, not that it has a value of infinity, because obviously it is finite, otherwise all circles (and anything made of circles) would be found infinitely large and it would just be a part of that vague concept called "infinity" and we wouldn't bother giving it the name 'Pi'.
Assuming you did mean infinitely long in terms of decimal places, if it has a finite number of decimal places, we will find out when we calculate the final decimal place. If not, then we would never know for sure. We do know that it has yet to be described as a divison of two rational numbers. Grade school teachers may use 22/7 (somewhat close for a few digits), I forget why...