Doubling the Square (Plato’s Meno)
To double a square’s area, how much longer should the side be? The answer lies in the diagonal — the puzzle Plato posed to a slave boy in the Meno.
Problem
Show, by cut-and-rotate, how many times larger the square built on a square’s diagonal is.
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 square of side s, area S. Cut along both diagonals into four pieces.
Why this approach works
Doubling the side would quadruple the area. Cut the square along both diagonals into four equal right triangles, then unfold them 180° outward over each edge: they surround the original square (left in the center), forming the square on the diagonal. New square = center + four pieces; the four pieces equal the original square, so the new one is exactly double. Its side is s√2, and (s√2)² = 2s².
Proof
Quartering the square along its diagonals and unfolding the pieces 180° outward builds the square on the diagonal around the original. The four pieces equal the original square, so the new one is exactly double — achieved by the diagonal (s√2), not by doubling the side.
Try it yourself
Check that the center square and the four unfolded pieces have equal area — the new square splits evenly in two. Hence “double.”
Related mathematics
Beyond MathVoyage
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