system
Given the following post, respond as if you're commenting on Reddit:
human
Is there an empirical method to calculate the surface area of an irregular solid?

Archimedes (apocryphally, at least) provided us with an empirical method of calculating the volume of an irregular solid using water displacement.

Is there any analogue for surface area? It would be nice if there were a method that doesn't rely too much on technology (I kinda want something that could've been produced by the Greeks), but I'm interested in any solution that doesn't use any kind of calculus / convergence of approximations arguments.

If there is no analogue for surface area, is there some kind of proof or heuristic argument that there can be no such method?
gpt
You could take a can of paint would form layer of certain thickness, submerge the object in thispaint, take it out, and see how much paint is missing. Divide by thickness and there you have surface area.

This would rely on the paint consistently forming layer of specific thickness, and while it's roughly true for all paints, it's not ever gonna be exact.

Mathematically, even if you had ideal paint, you also run into the problem caused by the shoreline paradox. You cannot compute shoreline of any real object. Similarly, surface area would be subject to changes as the thickness of your paint varies, ultimately requiring infinite amout of paint for paint that has no thickness.

Mathematically, this is best explained as shoreline having dimension higher than 1 but lower than 2. Length applies to 1-dimensional things, area to 2d objects. Similar to shoreline, any real object would not have 2d surface, it would be slightly nudging towards 3d with small cracks and such, not enough for it to be volume, but enough to cause shoreline paradox to be relevant.