It does not matter how big or small the circle is, π  always has the same value. This is why we call π  a mathematical constant.

We say that 3.14 is an approximation of π  because nobody has ever nor will ever calculate the exact value of π .  It is an irrational number. This means that the number has a never-ending and never-repeating string of digits after the decimal point.

The first known method for rigorously calculating better and better approximations of π  was developed by the Greek mathematician Archimedes of Syracuse in the third century B.C.E. Records suggest that the ancient Egyptians, Babylonians, and Indians all had fairly accurate approximations of π  before Archimedes’ work.

To date, we have calculated the first 62.6 trillion[1] digits of π . This is far more than we use for any practical purposes. NASA, for example, uses the first 15 or 16 digits to launch rockets and the first 40 digits would be enough for atom-sized accuracy even if the size of the circle was the size of the known universe.

By the way, here are the first 40 digits of π .

3.141592653589793238462643383279502884197

Pi is central to a great deal of mathematics. It crops up in all sorts of interesting places, not just those involving circles and spheres. One example is if you add up all the fractions where the numerator is one and the denominator is a perfect square, the answer turns out to involve π .

1+ 1/4 + 1 /9 + 1/16 + 1/25 +…+ 1/n² = π²/6 

More practically, π  is used in quantum mechanics (the study of sub-atomic particles), the study of light and other kinds of electromagnetic waves, and many kinds of engineering.