I did a project on proofs of the irrationality of pi during my undergrad. Strangely I didn't find anything that I would have considered "from the book"[0]. Even this "simple" proof is not really elementary and gets a lot longer if you have to spell it out.
In trying to find a proof with some explanatory value, like saying why pi is irrational the best I came across was Lindemann-Weierstrass[1].
In trying to find a proof with some explanatory value, like saying why pi is irrational the best I came across was Lindemann-Weierstrass[1].
[0] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
[1] https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_...