Plus even that isn't enough: 10/3 has an infinite decimal expansion (in base 10 at least) too, but if π = 10/3, you'd be able to find exact circumferences. Its irrationality is what makes it relevant to this joke.

A mathematician is also perfectly happy with answers like "4π" as exact.

Plus what's to stop you from having a rational circumference but irrational radius?

Writing this, I feel like I might have accidentally proved your point.