The Pythagorean Theorem Explained

The Pythagorean theorem is the most famous statement in mathematics. Stated simply:

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

In symbols: $a^2 + b^2 = c^2$, where $c$ is the length of the hypotenuse (the side opposite the right angle).

But like most famous things, it is both simpler and deeper than it first appears. Let’s go through what it actually says, why it’s true, and why — 2500 years later — it still matters.

What “square of a side” really means

The theorem is usually presented as a statement about lengths. But its earliest and most intuitive form is about areas.

Imagine a right triangle. Now draw a square on each of the three sides — each square pointing outward, with the side of the triangle as its base. You have three squares of different sizes.

The theorem says: the area of the square on the hypotenuse equals the combined area of the squares on the other two sides.

This is visual, almost physical. It is the form you find in Euclid’s Elements, Book I, Proposition 47. When you think of the theorem this way, $a^2$ isn’t just ”$a$ multiplied by itself” — it’s the literal area of a square of side $a$.

A proof you can draw

The simplest proof uses a picture. Take a square of side $a + b$. Arrange it two different ways:

Arrangement 1: Place four copies of the right triangle in the corners, with their legs of length $a$ and $b$ along the edges. What’s left in the middle? A tilted square whose side is exactly $c$ (the hypotenuse). The big square has area $(a+b)^2$, and four triangles take up area $4 \cdot \tfrac{1}{2}ab = 2ab$, leaving $(a+b)^2 - 2ab = a^2 + b^2$ for the middle square. So the area of the middle square is $a^2 + b^2$. But its side is $c$, so its area is also $c^2$. Therefore $a^2 + b^2 = c^2$. ∎

That’s it. You can draw it on a napkin.

The older-than-Pythagoras history

Pythagoras didn’t discover the theorem that bears his name. The relationship was known a thousand years earlier in Babylon. The clay tablet Plimpton 322, dating from about 1800 BCE, contains a list of Pythagorean triples — triples of integers $(a, b, c)$ satisfying the theorem — far longer and more sophisticated than what casual empirical observation would produce.

Egyptian builders knew that a rope knotted at lengths 3, 4, and 5 would form a right angle. Indian mathematicians knew general cases. Chinese mathematicians had a proof in the Zhoubi Suanjing (around 100 CE).

What the Pythagoreans — the followers of the philosopher Pythagoras of Samos — contributed was not the result but the proof: a general, logical argument that the relationship holds for every right triangle, not just for the ones anyone had tried. This commitment to proof is arguably the single most important invention in the history of mathematics.

Why it still matters

Almost every bit of technology you use depends on measuring distances, and almost every way of measuring distances descends from the Pythagorean theorem.

In two dimensions, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

— the theorem, applied to the coordinates. In three dimensions, add a $(\Delta z)^2$ under the square root. The same idea extends to any number of dimensions, giving the Euclidean norm of a vector in $\mathbb{R}^n$.

GPS systems, computer graphics, machine learning (cosine similarity), physics (Minkowski metric), statistics (least squares): all rest, at some level, on distances computed Pythagorean-style.

Where it fails

On a curved surface — the surface of the Earth, for instance — the theorem does not hold. Draw a triangle with one vertex at the North Pole and the other two on the equator, 90° of longitude apart. Each side is a quarter of a great circle. All three angles are right angles. Nothing Euclidean applies.

The generalization to curved spaces is non-Euclidean geometry, developed in the 19th century by Lobachevsky, Bolyai, and especially Riemann. Riemann’s geometry became the mathematical framework of Einstein’s general relativity. Pythagoras’s theorem is the simplest possible case of a much bigger story.

A problem that came from the theorem

The Pythagorean theorem says $a^2 + b^2 = c^2$ has many integer solutions — every Pythagorean triple. What about higher powers?

Does $a^3 + b^3 = c^3$ have positive integer solutions? What about $a^4 + b^4 = c^4$? In 1637, Pierre de Fermat wrote in the margin of his copy of Diophantus that no solution exists for any $n > 2$. He claimed to have a “truly marvelous proof” that the margin was too small to contain.

The proof — if Fermat really had one — was lost. The conjecture remained open for 358 years. It was proven by Andrew Wiles in 1994, using techniques Fermat could not possibly have imagined: modular forms, elliptic curves, Galois representations. The most famous unsolved margin note in history is now a solved margin note.

That, too, is part of the Pythagorean legacy.

Frequently asked

Is the Pythagorean theorem only for right triangles?

Yes, in its simplest form. For non-right triangles, the law of cosines generalizes it: c² = a² + b² − 2ab·cos(C). When C is 90°, cos(C) = 0 and we recover the Pythagorean theorem.

How do I prove the Pythagorean theorem?

There are hundreds of proofs. The most elementary uses a square of side a+b, arranged two different ways — once with four right triangles around a square of side c, and once with the triangles rearranged around two squares of side a and b. Comparing the two arrangements gives a² + b² = c².

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Formula

The Pythagorean Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The foundation stone of Euclidean geometry.

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Pythagoras

Greek philosopher and mathematician, founder of the Pythagorean school. Credited with the first general proof of the theorem that bears his name.

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Euclid

Greek mathematician active in Alexandria around 300 BCE. Author of the Elements, the most influential mathematics textbook in history.

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