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A dissection of a polygon $P$ is a partition of $P$ into a finite number of pieces, which can then be rearranged (via planar translations and rotations) and joined (without overlap) to form a new ...

I am looking for an attractive, but rigorous definition of area; say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...