Two squares, four triangles, and no algebra

Almost everyone meets $a^2 + b^2 = c^2$ as a thing to be remembered. It arrives as a formula, gets applied to a few triangles with convenient side lengths, and settles into memory somewhere near the quadratic formula — true, useful, and entirely opaque. Nothing about the symbols suggests why squares should be involved at all, or why a statement about areas should say anything about lengths.

The picture fixes that in about fifteen seconds.

Stated as a picture, the claim becomes concrete and slightly outrageous: cut up the two small squares, and the pieces will exactly fill the big one. Not approximately. Not for special triangles. For every right triangle there has ever been.

The argument #

Start with two identical squares, each with side $a + b$. Being identical, they have identical areas — that is the entire premise, and it is not in dispute.

Into each square go four copies of the same right triangle, the one with legs $a$ and $b$. In the first square they are arranged as a pinwheel, each triangle rotated a quarter turn from the last. In the second they are pushed into the corners in pairs, two of them forming a rectangle at the top and two at the bottom.

Both squares now contain exactly the same amount of triangle: four copies, $4 \cdot \tfrac{1}{2}ab$ of area, in each. So whatever space is left over must also be equal — the same total square, minus the same total triangle.

Now look at what is left over.

On the left, the four triangles leave a tilted square in the middle. Its sides are the hypotenuses of the four triangles, so each side has length $c$, and its area is $c^2$. (That the tilted region really is a square, and not merely a diamond, is worth one moment’s thought: at each of its corners, two of the triangle’s acute angles meet, and those two angles sum to $90°$ because the third angle of the triangle is a right angle. Four right angles, four equal sides — a square.)

On the right, the four triangles leave two rectangles that are in fact squares: one of side $a$, one of side $b$. Their combined area is $a^2 + b^2$.

Same starting square. Same four triangles removed. Therefore

$$a^2 + b^2 = c^2.$$

That is the whole proof. It uses no algebra beyond the observation that equals subtracted from equals leave equals, no similar triangles, no coordinates, and no trigonometry — which is fortunate, since trigonometry is built on top of this result and using it here would be circular.

What makes this convincing #

There is a temptation to treat a picture as an illustration of a proof rather than a proof in its own right — as though the real argument lives in symbols somewhere and the diagram is a courtesy to slower readers. That gets things backwards here. The diagram is the argument. Every step in it is checkable: the two outer squares are the same size, the eight triangles are congruent, the leftover regions are squares. Each of those claims can be verified by looking, and none of them depends on the particular $a$ and $b$ chosen for the drawing.

That last point is the one that does the real work, and it is also the one a picture is worst at communicating. A drawing necessarily commits to some specific triangle. What licenses the leap from this triangle to every triangle is that no step in the argument used anything about the drawn proportions — the pinwheel closes up for any $a$ and $b$, the corner-packing works for any $a$ and $b$, and the leftover regions are squares for any $a$ and $b$.

A skinny triangle, a nearly isosceles one, a triangle with irrational sides: the construction never cares. This is the sense in which a picture can be general, and it is why dissection proofs earn their place rather than merely decorating one.

A very crowded field #

The Pythagorean theorem may be the most-proved statement in mathematics. Elisha Scott Loomis spent decades collecting proofs and published 367 of them in The Pythagorean Proposition, sorted into families — algebraic, geometric, quaternionic, dynamic. The collection is not exhaustive and new ones still turn up.

Euclid’s own version, Proposition 47 of Book I, works quite differently. It drops a perpendicular from the right angle to the hypotenuse, splitting the large square into two rectangles, and then shows each rectangle equals one of the smaller squares by way of a pair of congruent triangles and a shearing argument. It is rigorous, it is beautiful — Euclid closes the Elements with the five regular solids and opens it with this — and it is considerably harder to hold in the head — mediaeval students called the diagram the bride’s chair, and the proposition itself acquired the nickname pons asinorum in some traditions, the bridge that donkeys could not cross.

The dissection above is much older in spirit. A diagram closely related to it appears in the Chinese Zhoubi Suanjing, where a square of side $7$ is decomposed around four $3$–$4$ right triangles, and a version of the rearrangement is traditionally associated with Bhāskara II in twelfth-century India, whose accompanying commentary is often rendered simply as Behold. Whether he wrote anything so terse is doubtful, but the sentiment is right: the diagram is doing the explaining, and prose added underneath it is mostly getting in the way.

Three hundred and sixty-seven is a suspicious number, though, and it is worth asking what it counts. A great many of Loomis’s entries are the same dissection with the pieces slid a different way, or the same algebra with the letters renamed, and the collection has no criterion for when two proofs are one. That is not a small quibble: how many proofs are there is only a question once when are two proofs the same has been answered, and it has not been.

The figures in this essay make the point concretely. The pinwheel arrangement, the corner-packed arrangement, the lopsided version and the trapezoid below are all drawn by a single piece of code with different arguments — one construction, four pictures. Anyone counting pictures would say four proofs; anyone counting constructions would say one. Neither answer is wrong, and the disagreement is entirely about where the boundary of a proof is drawn.

The honest distinctions are structural rather than visual. Euclid’s shearing argument is genuinely a different proof from the dissection, because it moves area by a different mechanism and needs a different set of prior results. A dissection with the pieces rotated a quarter turn is not. Loomis sorted his collection into four families for exactly this reason, and the families are a better count than the total — which suggests the real answer to how many proofs is nearer four than four hundred.

Not every proof is ancient. In 1876 James A. Garfield, five years before becoming president of the United States, published a proof using a trapezoid built from two copies of the triangle plus a third.

The trapezoid’s area can be had two ways. As a trapezoid, it is the average of the two parallel sides times the width: $\tfrac{1}{2}(a+b)(a+b)$. As three triangles, it is $\tfrac{1}{2}ab + \tfrac{1}{2}ab + \tfrac{1}{2}c^2$. Setting those equal and cancelling gives the theorem in two lines of algebra.

It is the same move as before — count one thing two ways — with the pleasing economy of needing three triangles rather than eight. Garfield published it in the New England Journal of Education, having apparently worked it out in conversation with other members of Congress, which is the most flattering thing anyone has ever been able to say about that institution.

The proof that was supposed to be impossible #

The remark above — that trigonometry cannot be used here because trigonometry is built on this theorem — was for a long time treated as settling a question rather than raising one. Loomis states it flatly in his collection: there can be no trigonometric proof of the Pythagorean theorem, because every trigonometric identity depends on it, so any such proof is circular. The claim stood essentially unchallenged for most of a century.

It is not quite right, and the reason is a nice piece of bookkeeping. Not every trigonometric fact descends from the theorem. The identity $\sin^2 + \cos^2 = 1$ certainly does — it is the theorem, on a triangle with hypotenuse 1. But the addition formulas, for the sine and cosine of a sum of angles, can be established from similar triangles alone, without ever invoking a squared length. Anything provable from those is fair game.

In 2023 Calcea Johnson and Ne’Kiya Jackson, then in their final year at St. Mary’s Academy in New Orleans, presented such a proof at an American Mathematical Society meeting, and published an expanded account with several more in 2024. The construction takes the right triangle, reflects it, extends the resulting figure into an infinite sequence of similar triangles, and sums the geometric series their sides produce. The answer to the sum is $c^2 = a^2 + b^2$, and the law of sines is the only trigonometric input.

Two features of this are worth separating. The mathematics is elegant but not revolutionary; a handful of non-circular trigonometric proofs were already known, Zimba’s from 2009 among them. What is genuinely notable is that a widely repeated impossibility claim about the most-proved theorem in mathematics turned out to be an unexamined assumption, and that it was examined by two secondary-school students working on a competition problem.

That is the useful lesson, and it points back at the picture. No proof of this kind can exist is a much stronger statement than no proof of this kind is known, and the first is a theorem requiring an argument while the second is a report on the literature. Loomis asserted the first and had grounds only for the second.

When the angle is not right #

The theorem’s converse holds too, and it is the reason the result is useful for building things rather than only for measuring them. If a triangle’s sides satisfy $a^2 + b^2 = c^2$, then the angle opposite $c$ must be a right angle. A rope with twelve evenly spaced knots, pulled taut into a $3$–$4$–$5$ triangle, produces a right angle with no instrument at all — which is a considerably older technology than the theorem that explains it.

And when the angle is not right, the equation fails in a predictable direction.

The law of cosines,

$$c^2 = a^2 + b^2 - 2ab\cos\gamma,$$

is exactly this statement with the error term written down. The correction $-2ab\cos\gamma$ is negative for an acute angle, positive for an obtuse one, and zero at exactly $90°$ — because that is where the cosine is zero. Setting $\gamma = 90°$ kills the term and returns the theorem.

Which reframes what the Pythagorean theorem is. It is not a fact about right triangles that happens to generalise; it is the general fact about all triangles, evaluated at the one angle where a whole term politely vanishes. Right angles are not special because triangles behave differently there. They are special because that is where the arithmetic gets easy — the same reason the exponential curve with base $e$ is singled out from all the others.

What the picture cannot show #

One thing the dissection genuinely does not establish: that the theorem depends on the geometry being flat.

Every step above quietly assumes that the angles of a triangle sum to $180°$, that squares exist, and that a figure can be moved without changing size. Those are properties of the Euclidean plane, and they fail elsewhere. On the surface of a sphere, a triangle with three right angles is perfectly ordinary, and $a^2 + b^2 = c^2$ is simply false — the correct relation involves cosines of the sides rather than the sides themselves. The picture cannot warn about this, because the paper it is drawn on is flat and has no way to represent its own flatness as an assumption.

It is worth putting a number on that, because “it fails on a sphere” sounds like a remark about exotic geometry and is in fact a fact about surveying. The spherical version is $\cos(c/R) = \cos(a/R)\cos(b/R)$, and expanding it for triangles small against the sphere gives

$$c^2 \approx a^2 + b^2 - \frac{a^2b^2}{3R^2},$$

so the correction is fourth order in the size of the triangle and inverse square in the radius of the world. On the Earth’s $6371$ km, a right triangle with legs of $1$ km has a hypotenuse the flat theorem gets right to well under a micrometre. Legs of $10$ km: wrong by $3$ millimetres. Legs of $100$ km: wrong by $2.9$ metres. Legs of $1000$ km: wrong by $2.9$ kilometres on a hypotenuse of $1414$.

That fourth power is why the flat theorem survived as an engineering tool for so long, and why it eventually stopped being one. A surveyor laying out a field is below the millimetre and can ignore all of this; a national triangulation spanning a few hundred kilometres cannot, and every geodetic survey since the eighteenth century has carried the correction. The angles tell the same story from the other side: the three angles of that $100$ km triangle sum to $180.0071°$, and the excess is measuring precisely the shortfall in $c$.

None of which is visible in the dissection, and none of it could be. The four triangles fit inside the square because the paper is flat, and a sheet of paper has no way to report its own curvature — the correction term contains $R$, and $R$ is the one quantity a drawing made on a surface cannot see from inside it.

That is worth holding onto as a general caution about diagrams. A picture shows what is true in the space it is drawn in, and is silent about which of its features came from the space rather than the argument.

The habit worth taking away #

The move that makes this proof work — count the same thing two different ways, then set the counts equal — is not a trick specific to triangles. It is one of the most productive habits in all of mathematics, and it turns up constantly: in stacking odd numbers into squares, where the same square is counted by rows and by shells; in Euclid’s algorithm drawn as a tiling, where the same rectangle is measured by two different units at once; and in essentially every combinatorial identity worth knowing.

The pictures differ. The move does not.

The ladder from here #

This essay is the first rung of a long ladder, and the rest of it is worth naming, because each rung is a different argument rather than a restatement:

Euclid’s own proof, drawn — the shearing argument that makes the bride’s chair diagram intelligible. The converse, and the knotted rope that used it to build right angles two thousand years before anyone proved it. Pythagorean triples, and why they are exactly the rational points on a circle. The tree that generates every triple once and only once. The theorem in three dimensions and then in $n$, where it becomes the definition of distance rather than a fact about triangles. What happens on a sphere, where it fails. The $p$-norms, where the exponent $2$ is replaced and the unit circle stops being round. And the crisis it caused, when the diagonal of the unit square turned out to be measurable by nothing at all.

The theorem is not a result. It is a doorway with about a dozen rooms behind it.