Determining the value of Pi empirically A shower thought hit me so hard I had to come here and ask:
Outside pure mathematics, where should I look, if I wanted to physically measure the best possible approximation for $\pi$? How many significant digits would I be able to get, assuming I had the best contemporary measuring equipment (and if necessary, fabrication facilities) available?
I was originally thinking planetary orbits, but actually measuring those to several significant digits seemed a bit difficult. Any other inverse-square force should also make pretty ellipses, and rotating things make nice circles and sine waves, but is there some especially handy thing or phenomenon that would give an exceptionally good approximation for $\pi$?
If at all possible, it would be nice to have an answer that takes all or most actual measurement problems (like possible disturbances, measurement errors, systematic errors, and the actual surroundings of the experiment) into account.
(Full disclosure: since it was a literal shower thought, and I've been thoroughly influenced by M. Hartl's propaganda, the original question in my mind was "does $\tau$ really exist")

Following the advice of user stafusa, I removed my request for help on improving this question so that we could get this question reopened. (I had tried to keep it constructive but admittedly it did show some unnecessary indignation.)
However, any such help would still be very much appreciated. Thanks in advance!
(I am not familiar with the meta question process, so if you think that would be the proper course of action, please feel free to open such a question.)
 A: Lazzarini's experiment with Buffon's needle is as far I know the only case where somebody has actually tried to "measure" $\pi$ empirically using a physical process (and there is a bit of a cheat/hoax involved). But in principle this could be used to measure the constant. As noted here, using longer needles can make convergence faster.
One can do this kind of thing with other random methods (even a version of the Monty Hall problem). Whether this counts as a physical measurement is somewhat arguable: certainly a physical process of experimentation is occurring, producing an empirical result, but the setup has been such that the result should approach $\pi$. Measuring 1/2 by flipping a coin is the same: the tricky part is getting a fair coin (experimental setup) - most likely the setup will have to be done by checking candidate coins for fairness, that is, ensuring that one gets a known result.
Still, in principle one could run through any physics formulary and get ways of estimating $\pi$. For example, an oscillating spring has period $T=2\pi\sqrt{m/k}$, so $\pi = T/2\sqrt{m/k}$. Planck's law for energy distribution of blackbodies gives $\pi = w(\lambda)\lambda^5 (e^{hc/\lambda kT}-1)/8hc$. So one could perform blackbody experiments and get approximations, although there has to be some care with the values of the ingoing constants - they have to be measured in a way that does not implicitly involve $\pi$.
A: Physics theories based on mathematics can't predict a value of $\pi$ different from the mathematics they're based on.  Physics experiments as you describe would rely on the underlying mathematical physics theory to determine a value for $\pi$, so it's a chicken and egg situation.
No numerical mathematics would work at all if $\pi$ were a different value so we're stuck with $\pi$ at the value it is.
Any experiment you devise which "measures" $\pi$ and returned anything but the correct value would simply tell you that the underlying physical theory is wrong, or the measurements are wrong or both.
And that's all they could tell you.
Measuring $\pi$ is no different from trying to measure 1 or 2 or any other number.
A: Just to set the history of pi in perspective

Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics.

....

Ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics, and to about five digits in Indian mathematics in the 5th century AD.

Another link:

The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation.
The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi.
The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71.
A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for pi, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

So certainly one could calculate pi by run of the mill methods.
A: Obviously very accurate values for Pi can be easily determined theoretically (I don't have a source but if I remember correctly the world record is something like 5 trillion decimal places), but if I understand your question correctly; you're asking about a way to experimentally measure the value of Pi. Presumably by measuring the circumference and radius of some circular object or path?
The first thing that comes to my mind would be a charged particle in a magnetic field, which moves in a circle according to the Lorentz force.
There are all kinds of applications in atomic and molecular physics that require very precisely determined magnetic fields that are isolated from external factors (such as the Earth's magnetic field etc.). I think you could easily achieve a very high level of precision and of course there's a lot of room to control the radius of the circle and thereby take measurements at different scales (think: giant LHC vs. practically any other, much smaller, particle accelerator).
You also don't experience any of the issues (like the one mentioned by @safesphere) associated with using massive planetary bodies on astronomical length scales.
