The thing is, with a lot of constants like pi, any measurements are attempting to capture a "pure" infinitely-precise value. And doing calculations in a purely abstract context can get a lot closer to that mythical value than measuring with even our most precise tooling. A constant like pi isn't a physical reference (like a gram or a meter). Rather, the physical thing approximates some abstract "perfect" value.

Think of it like this: we want to find the value of 1 + 1. So we create cubes that are 1 cm on each side, then we put two of them together and measure them. The measurement will not be exactly 2, it'll have some error (maybe 1.999896cm, or 2.010102cm). But instead of using physical measures, we can create imaginary cubes that are exactly, perfectly 1cm and two together is exactly 2cm.

So we can do the same thing with special constants like pi, e, and lots of others, and we can get much greater accuracy than by measuring real-world objects.