Misleading graphical proofs: the missing square paradox and more

The false mysteries of graphical proofs in mathematics

Discover how seemingly rigorous diagrams can fool us: the missing square paradox, the astonishing proof that every triangle is equilateral, and more.

Written by

DANIEL JUSTENS

PublishedOctober 7, 2025UpdatedSeptember 18, 20261 min read

The false mysteries of graphical proofs in mathematics

No one will dispute it: a good diagram can very often serve as a valuable aid to proving a theorem or a property. There are even cases where the diagrams in question actually contain a certain form of proof of the property.

The best-known example of this kind of situation is the Pythagorean theorem, for which the pair of diagrams opposite does indeed contain what can be regarded as a proof of the result. A well-constructed diagram can also lead, without further ado, straight to the angle-sum formulas for sine and cosine.

But how far can we trust a graphical representation? To what extent can our eye be fooled — or not — by an appearance that might be deceptive?

The missing square -----------------

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