How are first digits of $\pi$ found? Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either multiplication or division or any other operation on numbers.
Then how are the first digits of $\pi$ found -

3.1415926535897932384626433832795028841971693993...

In fact, more than 100,000 digits of $\pi$ are found (sources - 100,000 digits of $\pi$)
How is that possible? If these digits of $\pi$ are found, then it must be possible to compute $\pi$ with some operations. (I am aware of breaking of circle into infinite pieces method but that doesn't give accurate results.)
How are these digits of $\pi$ found accurately? Can it be possible for a square root of some number to be equal to $\pi$?

You make the assumption there is no way to calculate the digits of $\pi$. That is untrue; there exist many formulas to calculate the digits of $\pi$ (with their own proofs of correctness). One of the simplest (though very slow) formulas is the Leibniz formula for $\pi$. 
