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Answer:
There's a lot of them.
There are many different ways to calculate . The ones used by computers to generate tons of digits are usually infinite series.
The series that has been prominent in recent records for the most digits of pi is the Chudnovsky algorithm.
The algorithm is this:
For faster performance, it can be simplified to this:
Other algorithms have been used, but right now this is the one that is being used to set the recent records.
There are also some approximations that are used because they are very easy to calculate.
first, can be used to calculate a fairly accurate pi, but a better rational approximation is
This fraction is actually accurate to 6 digits and it is the best approximation of
in simplest form and with a denominator below 30,000.
There are several other approximations and if you want to learn more I would recommend looking at the Wikipedia page which has tons of algorithms for pi.