the simple answer is to tell "Jason Marshall" to go measure the diameter and circumference of a disk.
Step one, find a perfect natural disk of some macroscopic size. The Earth? They tried that in the original definition of the meter and botched the measurement and Newton told us that the Earth is not spherical. The Moon? Not on my budget. A sand dollar? Nope. A hydrogen atom's electron wave function? Neither macroscopic or circular -- the ground state wave function has a spherical symmetry but no hard edge.
Step two, give up and find a perfect manufactured disk. They don't exist. How about U.S. Coins? After all 31 U.S. Code § 5112 specifies what their mass and diameter
must be. But then 31 U.S. Code § 5113 ruins it by saying "The Secretary of the Treasury may prescribe reasonable manufacturing tolerances for specifications in section 5112 of this title" which is evidence that even the US Legislature recognized (at some point) that manufacture of a perfect disk is not without problems. Even then these are not disks, but approximate cylinders.
Step three, measure the circumference exactly. Well here we have another problem because no measurement has ever been exact.
Counting is exact, but measuring is estimating the ratio between one physical magnitude and another. So details matter. That the cylinder is curved makes the problem tougher and the procedure matters.
Mark the edge of the disk and roll it without slipping. How do you know you rolled it in a straight path without slipping?
Wrap a string/thread/chain about it, mark the overlap and measure the straightened string/thread/chain. How do you know you haven't stretched it out of shape by making it conform to the curve of the cylinder? Aren't you measuring how long a physical object is outside the cylinder, which is not exactly the same thing as the cylinder? Does the overlap point add to the length in some systematic way?
Step four, measure the diameter. How do you know your two points are a diameter?
Step five, take the ratio. This is either going to be a ratio of two rational quantities which is itself a rational number and therefore not pi, or its going to be a ratio of two estimated quantities (hopefully with error budgets) and thus it too will only be a (hopefully good) estimate of pi.
Say I have a disk which is between 779220779.20 and 779220779.25 carbon-carbon bond lengths in diameter and between 2447994275.50 and 2447994275.55 carbon-carbon bond lengths in circumference. Thus we estimate pi is between 9791977102/3116883117 and 48959885511/15584415584, which is a
good estimate, but lots of numbers are between those ratios.
2447994275.50 /779220779.25 = 9791977102/3116883117 < 1770721/563638 < 208341/66317 < pi < 312689/99532 < 417037/132747 < 48959885511/15584415584 = 2447994275.55/779220779.20
Another way to write this estimate is $$\tilde{\pi} = 3.141592653573 \pm 0.000000000133 = 3.141592653573(133)$$ which is in good agreement with the tabulated decimal value of pi, 3.141592653589793+.
Physicists know how important analysis of measurement errors are to the testing of physical theory, but since a circle has a mathematical definition, pure mathematics is of far greater utility in determining the 20th digit of pi than physical measurement.
$$\pi = 16 \tan^{-1} \frac{1}{5} - 4 \tan^{-1} \frac{1}{239} = \lim_{n\to\infty} \sum_{k=0}^{n} \frac{(-1)^k}{2k+1} \left( \frac{16}{5^{2k+1}} - \frac{4}{239^{2k+1}} \right)$$
This infinite sum only requires 8 terms to bound pi more tightly than the previous thought experiment on estimation.
pi = 3804/1195 - 72810068/1706489875 + 12476980230684/12184551018734375 -712697588470600764/24359780855939418203125 + 13569999650349009580148/14908446881486951862650390625 - 2325395850082757774745401724/78061994479096482923424854931640625 + 132828936352577206862383654376604/131742566878013892304309920002620849609375 - 2529107224465187544395498503569499028/72358338102298380214562374427593322601318359375 + ...
≈ 3.183263598326 - 0.042666569000 + 0.001023999999 - 0.000029257143 + 0.000000910222 - 0.000000029789 + 0.000000001008 - 0.000000000035 + ...