Q: How was the number π first discovered? How did we first figure out it was 3.14…?
Physicist: Unfortunately, the use of the number π is older than recorded history so… who knows? But the use of π at the beginning of recorded history wasn’t too sophisticated, so we can make guesses. If you want to predict how much water a cylindrical barrel can hold to within a couple percent, then you need to know π to at least a couple digits. All of this is to say that π, while full of mathematical mysteries or whatever, is not some abstract idea. Archimedes’ technique converges linearly to the true value of π (you gain about the same number of digits every time you use the algorithm).