In calculus there is a thing called Taylor Series which provides an easy way to calculate many irrational values to arbitrary precision.
Pi/4 = 1 – 1/3 + 1/5 – 1/7 + …
(from http://www.math.hmc.edu/funfacts/ffiles/30001.1-3.shtml )
Keep adding those terms until the number of digits of precision you want stabilize.
Taylor’s theorem is a powerful tool, but the derivation of this series using the theorem is beyond the scope of the question. It’s standard first-year university calculus and is easily googlable if you’re interested in more detail.
I didn’t mean to imply that this is the most practical method to calculate pi. That would depend on why you really need to do it. For practical purposes, you should just copy as many digits as you need from one of the many published versions. I was suggesting this as a simple introduction of how irrational values can be equated to infinite series.