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Why Does the Pythagorean Theorem Work?

Why Does the Pythagorean Theorem Work?

By Math Tools ·

Why Does the Pythagorean Theorem Work?

Almost everyone remembers a² + b² = c². Far fewer people remember why it's true. And yet the theorem may have more proofs than any other statement in mathematics. A 1940 collection by Elisha Loomis gathered 367 of them.

One was discovered by a future U.S. president. Another, long thought impossible, was found by two high school students in 2022. Let's look at why the theorem works, and why the squares in the formula are real squares, not just algebra.

It's a Statement About Area, Not Length

We usually use the theorem to find a length: if the legs are 3 and 4, the hypotenuse is 5. But the "squared" terms are literal. The theorem says:

For a right triangle, the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the other two sides.

For a 3-4-5 triangle: a 3 × 3 square (9) plus a 4 × 4 square (16) has the same total area as a 5 × 5 square (25). Once you think in areas, the proofs become pictures.

Proof 1: Rearranging Four Triangles

Take four copies of a right triangle with legs a and b and hypotenuse c. Arrange them inside a big square with side length a + b.

Arrangement 1: Place the triangles in the corners so their hypotenuses form a tilted square in the middle. The big square's area is:

(a + b)² = 4 × (½ab) + c²

Arrangement 2: Slide the same four triangles into two rectangles. The leftover space is now two squares, one a × a and one b × b:

(a + b)² = 4 × (½ab) + a² + b²

Same big square, same four triangles, so the leftover areas must be equal:

c² = a² + b²

That's the whole proof. It's often called the Chinese proof or the rearrangement proof, and a version of it appears in the ancient Chinese text Zhoubi Suanjing.

Proof 2: A President's Trapezoid

In 1876, Ohio congressman James A. Garfield, who would become the 20th U.S. president four years later, published a proof in the New England Journal of Education.

He used just two copies of the triangle plus one extra triangle, arranged into a trapezoid with parallel sides a and b and height a + b.

The trapezoid area formula gives:

Area = ½(a + b)(a + b) = ½(a + b)²

Adding up the three triangles inside it gives:

Area = ½ab + ½ab + ½c² = ab + ½c²

Set them equal and simplify:

½(a² + 2ab + b²) = ab + ½c²
a² + b² = c²

Garfield's proof is exactly half of the rearrangement picture. See the area formulas on the geometry formulas page.

Proof 3: Similar Triangles

Drop a perpendicular from the right angle to the hypotenuse. It splits the triangle into two smaller triangles, each similar to the original (same angles, different size).

If the perpendicular divides c into parts p and q (with p next to side a), similarity gives:

a / c = p / a   →   a² = cp
b / c = q / b   →   b² = cq

Add them:

a² + b² = cp + cq = c(p + q) = c × c = c²

Albert Einstein reportedly found a proof along these lines as a boy, around age 12.

An Insider Reference: The "Impossible" Trigonometry Proof

For generations, textbooks warned that you can't prove the Pythagorean theorem with trigonometry, because the core identity sin²θ + cos²θ = 1 is itself a consequence of the theorem. Using it would be circular reasoning.

In 2022, Calcea Johnson and Ne'Kiya Jackson, high school seniors at St. Mary's Academy in New Orleans, found a way around that trap. They built a proof using the law of sines and an infinite geometric series of triangles, without relying on sin²θ + cos²θ = 1. They presented it at an American Mathematical Society meeting in 2023, and in 2024 published it, along with further proofs, in The American Mathematical Monthly.

It's Much Older Than Pythagoras

Pythagoras (c. 570–495 BC) gets the credit, but the relationship was known long before him.

  • The Babylonian clay tablet Plimpton 322, from around 1800 BC, lists sets of numbers that fit a² + b² = c², such as (119, 120, 169).
  • Another tablet, YBC 7289, shows a square with its diagonal labeled with an approximation of √2 accurate to about six decimal places.

What the Greeks added was a general proof. Euclid gave one as Proposition 47 in Book I of the Elements.

The Converse Also Works

If a triangle's sides satisfy a² + b² = c², then it must have a right angle. Builders have long used this to lay out square corners: measure 3 units along one side, 4 along the other, and adjust until the diagonal is exactly 5.

Whole-number solutions like (3, 4, 5), (5, 12, 13) and (8, 15, 17) are called Pythagorean triples. There are infinitely many.

Two Concepts Worth Knowing

Similar Triangles

Two triangles are similar if their angles match. Their sides are then proportional. Similarity is the foundation of trigonometry and of Proof 3 above.

The Distance Formula

The distance between two points is the Pythagorean theorem in coordinates: d = √((x₂ − x₁)² + (y₂ − y₁)²). See the analytic geometry formulas.

Quick Answer: Why Is the Pythagorean Theorem True?

Arrange four copies of a right triangle inside a square of side a + b in two different ways. One arrangement leaves a square of area c² uncovered; the other leaves two squares of areas a² and b². Since the big square and the triangles are the same, c² = a² + b².

Try Them Yourself

Cut four identical right triangles from paper and try both arrangements inside a square. Seeing the leftover areas match is more convincing than any formula.

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