Either in terms of the exponential function, or by its Taylor series.

Oh yes, much simpler.

I guess you can get to "i" via algebraic equations, but linking it to a rotation unit outside a flat space seems tricky.

It's basic calculus. I'm really curious how you define the perimeter of a circle without basic calculus.

Of course you can define it, but it's hard to justify the specific definition without a flat geometry.

And you don't need proper calculus for the circumference, just the idea of limits.