Pythagorean Theorem: What It Says and the Formula

What the theorem says

The Pythagorean theorem applies to right triangles, the ones with a 90° angle. In its classic geometric form, the square built on the hypotenuse matches the sum of the squares built on the two legs. The same relationship, in standard algebraic notation, is written a²+b²=c². The legs are the two sides that form the right angle; the hypotenuse is the side opposite that angle, always the longest of the three. The geometric and algebraic forms say the same thing in different languages. The statement belongs to the framework of postulates and theorems described in the entry on Euclidean geometry.

Why it holds: the idea behind the proof

Wolfram MathWorld calls the Pythagorean theorem one of the most fundamental in geometry and notes that many different proofs exist, without giving an exact count. One direct way to see why the relationship holds starts from the squares in the geometric form itself: the one built on the hypotenuse has an area that can be cut into pieces and rearranged to match exactly the combined area of the two squares built on the legs, with no overlaps and no gaps left over. It is just one of the many routes to the same conclusion.

The same idea carries over outside the flat plane. Extend the reasoning to three dimensions and the space diagonal of a rectangular box can be found by applying the theorem twice: once on the base, to get the diagonal of the base rectangle, then again on a second right triangle built from that diagonal and the height. It is the same relationship, simply applied back to back — a construction that comes in handy for the shapes covered in the entry on solid geometry.

Solving for a missing side

Knowing any two sides of a right triangle, the third can always be found with the formula. For the hypotenuse, square each leg, add the results, then take the square root of the sum: c = square root of (a²+b²). For a leg, square the hypotenuse, subtract the square of the leg already known, then take the square root of what remains: a = square root of (c²-b²).

Worked example: a right triangle has legs measuring 3 and 4 meters. Squaring each one: 3²=9 and 4²=16. Adding them: 9+16=25. The square root of 25 is 5, so the hypotenuse measures 5 meters. It also works in reverse: if the hypotenuse is 5 and one leg is 3, the second leg comes from the square root of (25-9), which is 4.

The same calculation, applied to coordinates on a Cartesian plane, becomes the distance formula between two points: the difference between the x-coordinates and the difference between the y-coordinates act as the two legs, and the distance itself is the hypotenuse. That connection is described in the entry on analytic geometry, and the same right triangle is also the starting point for sine, cosine and tangent, covered in the entry on trigonometry.

The most common Pythagorean triples

Certain sets of three whole numbers satisfy a²+b²=c² without needing any square roots with decimals: these are called Pythagorean triples, and a triangle with sides in that ratio is a right triangle by construction, with no need to measure the angle.

The Plimpton 322 tablet, per the analysis by Neugebauer and Sachs cited by MacTutor History of Mathematics, is a list of whole-number Pythagorean triples. Neither 3-4-5 nor 5-12-13, the two most familiar triples, appears on it: the smallest one present is 45-60-75, a multiple of 3-4-5.

Content older than the name attached to it

The theorem carries Pythagoras’s name, but its content was known long before him. The Babylonian tablets analyzed by MacTutor History of Mathematics date to the Old Babylonian Empire, which flourished in Mesopotamia roughly between 1900 and 1600 BCE: according to this source, the Babylonians already knew the content of the theorem about a thousand years before Pythagoras.

Pythagoras was born on Samos, in Ionia, the son of Mnesarchus, in the first half of the 6th century BCE according to Treccani, a period MacTutor places more precisely around 570 BCE. He later moved to southern Italy, to Croton, perhaps around 529 BCE (Treccani uses “perhaps”: the date isn’t certain), and there founded a philosophical school that also became a religious community, aimed at the moral and political regeneration of its members. Within the school, the inner circle, the “mathematikoi,” lived permanently in the community: no personal possessions, a vegetarian diet. A second, outer circle, the “akousmatics,” lived in their own homes and joined the community only during the day.

On the question of who first proved it, sources stay cautious: MacTutor History of Mathematics writes that Pythagoras “may have been” the first to prove the theorem, a conditional that signals a guess rather than an established fact. It remains a traditional attribution, more than a documented one.