Wallace-Bolyai-Gerwien: Any Polygon Into Any Other

If two polygons have the same area, can you always cut one into a finite number of straight-edged pieces and rearrange them, with no gaps or overlaps, into the other shape exactly? The Wallace-Bolyai-Gerwien theorem says yes, always, for any two polygons of equal area — a triangle can be cut and reassembled into a square of the same area, an irregular pentagon into a rectangle, and so on, with the number of pieces always finite.

The proof works in two stages. First, any polygon can be cut into triangles (a standard triangulation), and any single triangle can be dissected into finitely many pieces that reassemble into a rectangle of the same area (cut a triangle's top off at half its height, rotate that piece, and it tiles into a rectangle). Second, any two rectangles of equal area can themselves be cut into pieces that reassemble into each other — by stacking thin congruent-width strips and shifting them, a rectangle of any given width can be sliced into another of any different width, same area. Chaining these steps means any polygon can be equidecomposed with a rectangle of matching area, and since equidecomposability is transitive, any two polygons of equal area are equidecomposable with each other, through that common rectangle.

The theorem is strictly two-dimensional: its three-dimensional analogue is false, which is exactly what Dehn's theorem (answering Hilbert's third problem) proved — some equal-volume polyhedra cannot be cut into finitely many pieces and reassembled into each other.