It was well known to the Ancient Greeks that the circle constant was pi, but they didn't know very well what it's value was, only the first couple digits. I found a way they could have known it to quite a bit of accuracy, scientifically.
You basically just measure how many drops of water completely cover the bottom of a cup with a certain radius , and divide it by the number of drops that cover the bottom of a cube with that radius. This works because out of a dropper the average volume of a drop stays the same. And it's easy to tell how many drops because you go until there are no dry spots.
It doesn't really matter what units of measurement you use because when you divide equations for a volume of a cup and a volume of a cube, the lengths drop out and you are left only with pi. You could start by finding the volume in drops of something small, and maybe find that it could hold a hundred drops until the bottom was completely wet. Then measure something larger with how many of those plus how many drops something larger holds. Work your way up to your cup and cube this way.
You could find that the cup would hold 1227184.6 drops and the cube 390625 which would give pi as 3.14159104 which is accurate to 6 digits, this would correspond to a 25 inch cylinder and cube dish. But you probably wouldn't be that accurate, Suppose you were off by 20 drops for each, you would still get 3.14138 or 3.1418 depending on which way you were off. And you can do the experiment any number of times and take the average to improve precision.