One of the most fundamental relationships in geometry, connecting the three sides of every right triangle.
In any right triangle — a triangle with one 90° angle — the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Over the centuries, mathematicians have discovered more than 370 distinct proofs. Here are three of the most celebrated.
Arrange four identical right triangles inside a large square. The area left over in the centre equals c². Rearranging them proves a² + b² = c².
Drop a perpendicular from the right angle to the hypotenuse. The two smaller triangles are similar to each other and to the original, yielding the relation directly.
Place the same four triangles differently inside the same square. Comparing the two area expressions — (a+b)² — gives a² + b² = c² algebraically.
Fill in any two sides and the missing one will be calculated instantly. Leave the side you want to find blank.
Only right triangles. All values must be positive.
Builders use the 3-4-5 rule (a classic Pythagorean triple) to ensure walls and foundations meet at perfect right angles.
Distances between coordinates on a flat plane are computed with the theorem. GPS systems generalise it into three dimensions.
Every distance calculation in 2-D and 3-D engines — collision detection, lighting, camera transforms — relies on the theorem.
Vector magnitudes, signal processing, and structural stress analysis all use the Euclidean distance formula derived from it.
Babylonians — Clay tablet Plimpton 322 records Pythagorean triples centuries before Pythagoras was born, showing the relationship was known in Mesopotamia.
Baudhayana (India) — The Sulba Sutras state the theorem for rectangles and its use in altar construction.
Pythagoras of Samos — The Greek philosopher is traditionally credited with the first general proof, giving the theorem its name.
Euclid — Included a rigorous proof as Proposition 47 of Book I in his monumental work Elements.
James A. Garfield — The future 20th U.S. President published an original trapezoid-based proof in the New England Journal of Education.
The Pythagorean theorem works only when one angle is exactly 90°. The Law of Cosines extends the same idea to any triangle, whatever its angles.
Place vertex C at the origin, B at (a, 0) along the x-axis, and A at (b cos C, b sin C) — the exact point reached by travelling distance b from C at angle C. Side c is then the straight-line distance from A to B.
Apply the distance formula c² = (xA − xB)² + (yA − yB)², then expand and simplify:
When the angle C is 90°, we have cos(C) = 0. The law of cosines therefore simplifies to the Pythagorean theorem: