How do you formalize things like perimeter and area?

After taking geometry, I realized I didn’t really know how to formally define perimeter and area. I have the feeling that at least for the kind of shapes basic geometry tends to deal with, there wouldn’t be any significant pathological behaviors that would make it really difficult, but I don’t really know how one would do it. In particular, while working with polygons seems easy enough, jumping to round figures like circles seems to be where the difficulty lies for me.

I’ve seen a few different approaches, but I haven’t really looked deeply into them, so I would be interested if someone could tell me which is the “best” way to formalize area (I would guess perimeter and volume follow the same idea); that is, it can be used to rigorously define area and prove their basic theorems from Euclidean geometry: things like area of a triangle, circumference and area of a circle, invariance under rigid motions and dissections, say.

These are some of the things I’ve seen:

  1. Synthetic. In Hartshorne’s Geometry: Euclid and Beyond, he tries to formalize the notion of area discussed in Euclid. He sticks to polygons, as that’s mostly what Euclid covered. He defines a “figure” as a subset of the plane that can be decomposed into triangles and shows that there is a unique definition of “area” such that: every triangle has a positive area; congruent triangles have equal area; the area of a union of non-overlapping figures is equal to the sum of their areas; the area of a triangle is bh/2. While Hartshorne mentions circles and other round figures, he doesn’t spell out a way to extend the notion of area to such shapes.
  2. Calculus. Hartshorne does mention that the modern approach would be to define the notion of area via the definite integral, especially since the Cartesian plane is essentially the unique model of Euclidean geometry. I can see how this would be done, and it makes sense – perimeter would be arc length, volume would be triple integrals, etc. Is this the modern way of formalizing area? A difficulty I see would be translating questions about rigid motions and dissections into this context.
  3. Measures. On the Wikipedia page about area, it seems to take a similar approach to Hartshorne; that is, showing there is a unique function (Jordan measure) from a set of “figures” in the plane (here, measurable sets) to the reals satisfying a set of axioms (nonnegative, respects set difference, respects congruence, etc.), though these are based on unions of rectangles. The problem I see with this is that it seems difficult to actually do area calculations on anything not a rectangle, unless you used something like a Riemann sum.

Are either of these the “right” way of formalizing the notion of area (and perimeter and volume and such)? That is, using one of these, can one rigorously build up to the standard theorems found in geometry for both polygons and round shapes?

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