Concept
In a right triangle, the square of the hypotenuse equals the sum of the other two squares. Related calculations and proofs appear in Babylonian, Indian, Chinese, and Greek sources.
Understand it in one breath
The sum of the areas of the squares on the legs of a right triangle equals the area of the square on its hypotenuse. Related number lists and geometric rules survive in Mesopotamian, Indian, Chinese, and Greek traditions, so the result should not be reduced to one discoverer. Euclid's Elements contains a general proof, and the relation remains a standard tool for Euclidean distance.
At a glance
Key formula
Worked examples
- 1
Q.a=3, b=4 → c=?
- 2
Q.a=5, b=12 → c=?
Key moments
A Babylonian table of right triangles
The Babylonian clay tablet Plimpton 322 records number patterns related to Pythagorean triples such as 3–4–5 and 5–12–13, long before a surviving general proof.
A general proof in the Pythagorean tradition
In the Greek mathematical tradition, the relationship was established for every right triangle rather than only for numerical examples — a high point of the Pythagorean belief that number orders the world.
Incommensurability — a length no fraction can capture
For a square of side 1, the diagonal is √2, which is not a ratio of integers. The result emerged within the Pythagorean tradition, while the discoverer and circumstances remain uncertain.
Descartes and coordinate geometry
With the coordinate plane, the distance formula between (x₁,y₁) and (x₂,y₂) became the Pythagorean theorem written in algebraic language.
Modern applications
GPS coordinate calculations, distance in computer graphics, Euclidean distance in machine learning, and signal norms — the theorem is evaluated thousands of times in each rendered frame.
Beyond MathVoyage
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