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There is a mathematical constant $\pi$ that is known to a gazillion decimal places. Based on the usual axioms of geometry this value is coincident with the ratio of the circumference of a circle to its diameter. Are there any large-scale experiments to confirm the ratio of a circle's circumference to its diameter?

For example, there have been many experiments to measure the speed of light in a vacuum $c$. There are ongoing experimental efforts to measure $c$ with ever increasing precision. Are there similar efforts focused on measuring the ratio of the circumference of a circle to its diameter and comparing this to the computed values for $\pi$?

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    $\begingroup$ +1 This is a perfectly valid question. It's a measurement that can in principle be made -- and the answer is that, if your measurement is found to deviate from the theoretical value, it's because space is curved! If $\pi$ is smaller than expected, space is curved like the surface of a sphere; if $\pi$ is greater than expected, space is curved like a saddle or potato chip. Of course, general relativity says that space is curved, as is spacetime on the whole. But the effect on what you would measure for $\pi$ is insanely small, except in the most extremely curved regions of spacetime. $\endgroup$
    – Adam Herbst
    Commented Feb 17, 2023 at 4:18
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    $\begingroup$ However, the ratio of circumference to (geodesic) diameter is not actually called $\pi$ in curved space. The symbol $\pi$ is really reserved for the value of that ratio in flat space only, afaik. $\endgroup$
    – Adam Herbst
    Commented Feb 17, 2023 at 4:21
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    $\begingroup$ Gauss experimentally measured the value of $\pi$ by using surveying techniques to measure the sum of the angles of a triangle with vertices at mountain peaks located roughly 100km apart. It seems to be unclear whether his primary purpose in this experiment was to verify the value of $\pi$. Not long afterward, Lobachevski designed an experiment specifically designed to test the value of $\pi$ using a triangle with vertices at Sirius and the two locations of the earth six months apart. If the question is open, I will post this as an answer, with more details. $\endgroup$
    – WillO
    Commented Feb 17, 2023 at 5:00
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    $\begingroup$ I should add that "measuring the value of $\pi$" is a poor way to phrase this, because the value of $\pi$ is, by definition, a definite integral of the arctangent function, and its value is fixed by that definition just as surely as the value of the number 7 is. You are writing $\pi$ to mean something like "the ratio of the circumference to the diameter of a physical circle" or "the sum of the angles of a physical triangle". It would be better to use some symbol other than $\pi$ to represent that thing, but in my comment above I stuck with your notation. $\endgroup$
    – WillO
    Commented Feb 17, 2023 at 5:03
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    $\begingroup$ @Him: Of course you are free to define $\pi$ in any of many equivalent ways, and logically it doesn't matter which you choose. But it strikes me as quite odd and unmotivated to define it via the $\zeta$-function, because the fact that $\zeta(2)=\pi^2/6$ comes as a surprise to just about everybody who sees it for the first time, which means that most people already have a definition of $\pi$ in mind before they become aware of this equality. $\endgroup$
    – WillO
    Commented Feb 17, 2023 at 5:22

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