Thank you for the kind words.

As to your question, I assume you know that Pi is in fact transcendental, which is a stronger property than being irrational. I have a page with a reference to Lindemann's theorem from which transcendentality of Pi follows with the help of Euler's formula:

https://www.cut-the-knot.com/impossible/sq_circle.shtml

However, you may also want a proof of an apparently simpler result, i.e. a direct proof that Pi is irrational. Lambert proved this in 1767 and then Legendre came up with another proof in 1794. At the same time Legendre also proved irrationality of Pi^2. This is mentioned in "A History of Pi" by P. Beckmann. So this are the names to look for if you are going to search the Web.

Gauss estimated Pi by counting grid points inside a circle. In the present day terminology this probably belong to the geometric number theory.

All the best,
Alexander Bogomolny