Is there a difference between the calculated value of Pi and the measured value? The mathematical value of Pi has been calculated to a ridiculous degree of precision using mathematical methods, but to what degree of precision has anyone actually measured the value of Pi (or at least the ratio of diameter to circumference), by actually drawing a circle and then measuring the diameter and circumference?
If these two values differ, is the resulting difference (discounting inaccuracy in measurement) the result of the curvature of the surface on which the circle is drawn, or in the case of a circle in space in zero gravity (as much as that can exist), the curvature of space-time?


Taking pi to 39 digits allows you to measure the circumference of the
  observable universe to within the width of a single hydrogen atom.

http://gizmodo.com/5985858/how-many-digits-of-pi-do-you-really-need
Even if we went out and measured pi with the most sophisticated instruments that we have, we wouldn't be able to get very many digits of accuracy. Way fewer than 39 digits. 
How many digits of pi do physicists actually need or use?

... in practice, π≈3.141592654 would be OK everywhere in the part of
  physics that is testable.

and 

16 digits, for converting frequencies from Hz to angular frequency.
  Frequencies can now be measured with a precision approaching 1 part in
  10^16, so dealing with those numbers would require knowing Pi to 16
  digits or so.

https://physics.stackexchange.com/questions/9621/how-many-digits-of-pi-are-required-in-physics
As others have pointed out, we don't really get the value of pi experimentally. Pi is a particular real number that we can calculate in various ways not having anything to do with circles. And our ability to calculate decimal digits $\pi$ goes into the trillions; while physically the best we could possibly do would be in the low teens. 
Here's a page full of closed-form expressions for $\pi$ that show how this particular real number arises in all sorts of ways that have nothing to do with circles.
And here are five trillion digits of $\pi$.
