Would it make sense to talk about approximated constructions in euclidean geometry?

Question

I got curious with something: I know that in euclidean geometry we talk about constructible and non-constructible structures, do we have the concept of approximation in euclidean geometry? I mean, we can't construct $\pi$, but we can have an approximation with the method of exhaustion - but I don't know if this method belongs to the domains of euclidean geometry, I've heard about it only on calculus.

Answer 1

The distinction between geometry and calculus in this context strikes me as artificial; problems do not always fall into rigidly (or even not-so-rigidly) defined areas of study.

To answer your question: yes, it is meaningful, geometrically, to say "this square has the same area as this a circle to within 0.0001%", or whatever precision you want. As Jack said in the comments, you can construct lengths of arbitrary rationals, and so arbitrary approximations of reals.

A more geometric construction that doesn't "know" that $\pi\approx 3.14\dots\,$ might look like this: say you had some 1000-gon or something and you wanted to say its area was close to the area of a circle. To get a bound on the error, you would simply construct the dual 1000-gon and then calculate the difference between them (which could be done purely geometrically).