Two polygons are congruent by dissection iff they have the same area. In particular, any polygon is congruent by dissection to a square of the same area. Laczkovich (1988) also proved that a circle is congruent by dissection to a square (furthermore, the dissection can be accomplished using translations only).
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ReferencesBogomolny, A. "Wallace-Bolyai-Gerwien Theorem." http://www.cut-the-knot.org/do_you_know/Bolyai.shtml.Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., pp. 50-51, 1991.Laczkovich, M. "Von Neumann's Paradox with Translation." Fund. Math. 131, 1-12, 1988.
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Cite this as:
Weisstein, Eric W. "Wallace-Bolyai-Gerwien Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Wallace-Bolyai-GerwienTheorem.html