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?
 A: There's an underlying error in the question, namely the assumption that being in a curved space would result in a "different measured value of $\pi$".
What happens in a curved space is that the ratio between a circle's circumference and diameter is no longer the same for all circles. More precisely, the ratio will depend on the size of the circle. For small circles (with diameter tending towards 0) the ratio will converge towards the one unchanging mathematical constant $\pi$ -- as circles get larger the ratio will either become larger and smaller according to whether the curvature of space is negative or positive.
However, $\pi$ as the limit of $\frac{\text{circumference}}{\text{diameter}}$ for small circles is the same mathematical constant for all possible curvatures of space.
According to the General Theory of Relativity we live in a slightly curved space. This has been measured directly in the vicinity of Earth by the Gravity Probe B experiment. The experiment didn't actually measure the circumference of a large circle, but the results imply that the geometric circumference of a circle approximating the satellite's orbit around the earth is about one inch shorter than $\pi$ times its diameter, corresponding to $\frac CD\approx 0.9999999984\, \pi$. (The curvature is caused by Earth's mass being inside the orbital circle. A circle of the same size located in empty space would have a $\frac CD$ much closer to $\pi$).

Science fiction authors sometimes get this wrong. For example in Greg Bear's Eon there's a mathematician character who concludes she's in a curved area of space by measuring the value of $\pi$ and getting a nonstandard value. I headdesked -- it doesn't work that way.
A: 
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 $\pi$.
A: 
By definition, pi is the ratio of the circumference of a circle to its diameter.

So how can others answer that this value is not best found from the most precise measurement we can make?
In the real world, a perfect circle has a definite value for radius. Pi is thought to be a never ending, non repeating decimal. This kind of value cannot truly represent a real perfect circle. This kind of value can only be used for approximations. The formulas found in the link of user4894's post do just that, approximations.

Atomic-force microscopy (AFM) or scanning-force Microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit.
  AFM - wikipedia

We should really use this to measure a circle! But how would we make a perfect circle to measure?
Circles are abstract, it might not even be possible to make a perfect circle to measure. I cannot find instances of anyone trying to measure a circle with AFM. But C = Pi*D still stands as the definition for pi that all those fancy calculations attempt to approximate. Good question OP!
One good way to approximate the right dimensions for C and D is to fit a polygon inside a circle, and keep subdividing it until you reach an absurd amount of vertices. This polygons perimiter is a good approximation of the circumference of the circle, take the distance between two points that are straight across from one another for diameter. We can reach the highest levels of precision with this method, althrough there are more efficient ways to generate good pi values, it seems the most trustable to me. Very simple.
