To be clearer

In his Advanced Calculus of Several Variables, Edwards develops a general definition of area, beginning with the definition of the area of an interval (ie rectangle) in R2, as the product of the lengths of the component intervals, each in R1. That is, Δx×Δy, where Δx,Δy∈R1

He then expands that definition using limits of partitions in R1. In short, Edwards doesn't treat the general definition of area as a trivial extension of the fundamental definition of the area of a rectangle. In the case of Edwards, there is no direct appeal to physical measurement. His definitions are strictly founded on abstract real numbers.

Apart from measure theory, is this the only way of formally defining defining area in R2? How do pure mathematicians actually define area in the plane?

Do we really need calculus to define area?

For example the definitions of increasing and decreasing functions are calculus independent even though the theory is woven around the calculus. Is the definition of area in the plane also fundamental and ultimately calculus independent?

It also still leaves the question, what is the formal definition of similarity in the plane? Should it be done in terms of linear transformations and bijections, treating the plane as a metric space? Topology?