As noted in other comments, any two equal-area polygons admit such a subdivision. This shows that the definition of area for plane polygons in principle requires nothing more than the formula for the area of a square (which is obvious by similarity arguments) and a finite geometric transformation from an arbitrary polygon into a square. Surprisingly, a similar statement is false for polyhedra, e.g a regular tetrahedron and a cube of equal volumes cannot be subdivided into a finite number of common pieces. There is another obstruction, called Dehn's invariant. This shows that the definition of volume really requires the full power of measure theory even for the simplest cases.