While Pi has been calculated to more than a trillion (10^12) digits, you’ll never need that many in the real world.
For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom. [emphasis mine]
I think 39 decimal places is more than enough for any real world applications, at least for now.
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[Source: Wikipedia: Pi, retrieved 3/14/12]