You don't need any geometry to define pi. At uni, I had it defined as the smallest positive zero of sin(x). And sin(x) was defined axiomatically (in terms of re

You don't need any geometry to define pi. At uni, I had it defined as the smallest positive zero of sin(x). And sin(x) was defined axiomatically (in terms of real analysis) in conjunction with cos(x) (and then shown to be equal to its Maclaurin expansion).