The Pythagorean Theorem

The Pythagorean theorem is probably the most famous result in all of mathematics. In a right triangle — a triangle with one 90° angle — the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Symbolically:

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

where $c$ is the hypotenuse and $a$, $b$ are the two legs.

The geometric picture

The theorem is usually taught as a statement about the lengths of sides, but its geometric meaning is about areas. If you draw a square on each of the three sides of the triangle, the area of the square on the hypotenuse equals the combined area of the other two squares. This visual form is how Euclid proved it in Book I of the Elements around 300 BCE.

Who discovered it

Tradition credits Pythagoras of Samos (ca. 570–495 BCE) and his followers, the Pythagoreans. But the relationship was already known to Babylonian mathematicians around 1800 BCE — the clay tablet Plimpton 322 lists dozens of what we would now call Pythagorean triples, triples of integers $(a, b, c)$ that satisfy the theorem. The Egyptians used the 3-4-5 right triangle for surveying, and similar knowledge existed in ancient India and China.

What the Pythagoreans are credited with is the first general proof — the step from “this works for the triangles we measured” to “this works for every right triangle.”

Why it matters

The theorem is the backbone of distance measurement in Euclidean space. In two dimensions, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

— a direct application of the theorem. In three dimensions it extends to $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$, and the same idea generalizes to $n$ dimensions, giving the Euclidean norm of a vector. Without Pythagoras there is no trigonometry, no analytic geometry, and essentially no modern physics.

Beyond Euclidean space

The theorem only holds on a flat plane. On a curved surface — the surface of a sphere, for instance — the “squares” on the sides do not add up the same way. Non-Euclidean geometry, developed in the 19th century by Lobachevsky, Bolyai, and Riemann, generalizes the idea to curved spaces and underlies Einstein’s theory of general relativity.

Pythagorean triples

Triples of integers satisfying $a^2 + b^2 = c^2$ are called Pythagorean triples. The simplest is $(3, 4, 5)$; others include $(5, 12, 13)$ and $(8, 15, 17)$. There are infinitely many, and they can all be generated from a single formula discovered by Euclid. The problem of finding triples where the same relationship holds with higher powers — $a^n + b^n = c^n$ for $n > 2$ — is Fermat’s Last Theorem, famously proved only in 1994 by Andrew Wiles.

Leg a

3.00

Leg b

4.00

a²

9.00

b²

16.00

a² + b²

25.00

c²

25.00

c = √(a² + b²)

5.00

The blue and orange squares (a² and b²) together always have exactly the same total area as the gray square on the hypotenuse (c²) — this is the Pythagorean theorem stated as an area relationship.

Frequently asked

Who discovered the Pythagorean theorem?

The theorem is named after the Greek mathematician Pythagoras (ca. 570–495 BCE), but the relationship was known to the Babylonians and Egyptians a thousand years earlier. Pythagoras and his school are traditionally credited with the first general proof.

Why is the Pythagorean theorem so important?

It is the cornerstone of Euclidean geometry, the basis for computing distances in any number of dimensions, and the starting point for trigonometry, analytic geometry, and countless applications in physics and engineering.

How many proofs exist?

Hundreds. Elisha Scott Loomis collected 367 distinct proofs in his 1927 book. There are geometric, algebraic, and even vector-based demonstrations, and new ones are still being found.

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Formula

Euler's Polyhedron Formula

For any convex polyhedron, vertices minus edges plus faces equals two. A simple relationship that launched topology.

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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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