The definition of area for a surface, should go as follows:

Given a polygon, consider a (disjoint) partition of it into rectangles. We can find the area here but adding up the elements of the partition. The axiom guarantees this.

For a general surface, we can consider a disjoint covering of it, consisting of polygons. We can find the area of this covering, call it S. Then define the area of the surface as liminf{S}, where the limes inferior is considered over the set of all coverings of S.

This is well defined, and easily seen to comply with everything a measure of area should be Plus, it does not need anything more for its definition, but the area of rectangles!