system
Given the following post, respond as if you're commenting on Reddit:
human
If we don't know the exact value for Pi, how can we know the exact area of a circle if that is 2(pi) Rad?
gpt
We *do* know the exact value of pi. It's... pi. You are most likely conflating "knowing an exact value" with "having a terminating or repeating decimal representation". A number that lacks the latter is no less exact than any other number. Also, we may compute the decimal representation of pi for an arbitrarily large number of decimal places if we really wanted. There's rarely any point though.

(Finally, note that we may show that in planar Euclidean geometry, the ratio of circumference to radius is the same for all circles. So we may *define* 2\*pi to be that number such that C/r = 2\*pi. So whether you were able to calculate a decimal representation of pi or not, we know that C = 2\*pi\*r because we have defined it to be that.)