This is being discussed in micromass's "Math stuff that hasn't been proven" thread, but I want to be particular about this topic. Essentially, I think I'm looking for a "proof" or derivation of pi from few first principles. Honestly I have no idea which is the "purest", most motivating question to ask that reveals the number [itex]\pi[/itex] as a constant. After defining some (relatively) intuitive axioms, I think it's okay to forsake some rigor (proofs of existence/uniqueness) so that we can really understand the meat of the origins of [itex]\pi[/itex]. Probably, a few first questions are: What is a circle? Why is the ratio of a circle's circumference to its diameter a constant; moreover, why is it a number between 3 and 4? Which "first principles" should we accept for the derivation of this number? How can we use a limiting expression to evaluate a decimal approximation for pi? The goal is to carefully illustrate how we can begin with definitions, and arrive at a useful definition and approximation for pi. To make analogy: teaching elementary calculus a few weeks ago, I tried to develop the use of e. I'm pretty much looking for a similar derivation of pi. Obviously it would rely more on a geometrically motivated problem, instead of a beginner's calculus one. Here is a quick recap of how I did this. Sorry for tl;dr.