Rectangle into a Square — Equal Area (Quadrature)
A rectangle and a square of equal area. Two cuts turn the rectangle into the square — the heart of “quadrature,” measuring area by a square for two millennia.
Problem
Cut and rearrange a p×q rectangle into a square of equal area (side √(pq)). Example: 9 × 4 → 6 × 6.
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
A p×q rectangle (e.g. 9×4). Turn it into an equal-area square (side √(pq)=6).
Why this approach works
Overlap the rectangle and square at the bottom-left. The part where the rectangle juts out to the side equals (in area) the part where the square rises higher. So cut the jutting piece and lift it to fill the top. For 9 × 4, the square is 6 × 6; cut the 3 × 4 overhang into two 3 × 2 pieces and raise them into the 6 × 2 gap — a square. Area is preserved, so the side is exactly √(pq).
Proof
Overlapping the rectangle and square, the side overhang is cut and lifted into the top gap, forming the square. Area is preserved, so its side is √(pq).
Try it yourself
Check that the overhang (3×4=12) equals the top gap (6×2=12) — forced by equal total area, so it fits exactly.
Related mathematics
Beyond MathVoyage
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