Axiom:Area Axioms

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Axioms of Area

When discussing areas of surfaces, we usually take the following statements to be axiomatic:

$(1) \quad$ The area of a square with a side of length one unit is defined to be one square unit.
$(2) \quad$ Let a surface be divided into a finite number of smaller non-overlapping surfaces.
Let the smaller surfaces cover the entire larger surface.
Then the sum of the areas of the smaller surfaces equals the area of the larger surface.
$(3) \quad$ Equal surfaces have equal areas.