Geometry? Pi is an infinite series, as are the trigonometric functions. How can you explain the circumference of a circle without invoking pi?
Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
You can truncate the series at an appropriate place of your choosing. But yes, \Pi as a number does not even exist unless you add the Reals, so that would need infinity. What I am saying is, one can work with the Rationals.