Wallace–Bolyai–Gerwien theorem

Statement

Let $P$ and $Q$ be two simple polygons in the Euclidean plane with the same area. Then $P$ can be cut into finitely many polygonal pieces that can be moved by isometries to form $Q$ , with disjoint interiors.

Remarks

The proof cuts a polygon into triangles. Each triangle can be cut and reassembled into a rectangle, and each rectangle into a rectangle with one side of length $1$ . Stacking these rectangles gives a rectangle of width $1$ and height equal to the area. So $P$ and $Q$ can both be cut into the same rectangle, and the two dissections combine, since the relation "can be cut into" is transitive. The analogue in dimension $3$ is false: this is the negative answer to Hilbert's third problem.

Going deeper

For a student

The theorem justifies the old method of finding areas by cutting and rearranging. An example is the proof that a parallelogram has the area of a rectangle with the same base and height. It shows that in the plane, equal area is the same as scissors congruence. In space there is a second invariant, the Dehn invariant, and equal volume is not enough.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Euclidean isometry concept
Polygon concept
Wallace–Bolyai–Gerwien theorem statement

Tag	`5RQGCEQYK5`
URI	`m2://5RQGCEQYK5`
Kind	statement, theorem
Writing	W0 stub
MSC	52B45, 51M25
Status	This copy of the site has no build data. Run `m2 status 5RQGCEQYK5`.
About	Polygon
Uses	Polygon, Euclidean isometry
Same as	Wikidata Q834211 Wikipedia Wallace–Bolyai–Gerwien theorem