The Bolyai-Gerwien theorem

The Bolyai-Gerwien theorem states that any two two-dimensional shapes with the same area can be transformed into each other through a finite series of cut-and-paste operations using only straight lines. Essentially, if two polygons have equal area, you can cut one into finitely many pieces and reassemble them without gaps or overlaps to form the other shape. This theorem highlights that area is a complete measure of the shape’s size in the plane, and all shapes of equal area are fundamentally equivalent in terms of how they can be dissected and reconfigured.