What is the Pythagorean theorem? a² + b² = c², explained

The Pythagorean theorem says that in any right triangle, one with a 90-degree corner, the square of the longest side equals the sum of the squares of the other two: a² + b² = c². The longest side, c, is called the hypotenuse and sits opposite the right angle. If the two shorter sides are 3 and 4, then 9 + 16 = 25, so the hypotenuse is 5. It is how builders square a corner and how computers measure distance.

In short

• For a right triangle with shorter sides a and b and longest side c, a² + b² = c², and the reverse also holds: if the squares add up, the corner is a right angle.
• The squares are real areas: the square built on the hypotenuse has exactly the area of the two smaller squares together.
• Babylonian scribes used the rule more than a thousand years before Pythagoras, and the same idea gives distance in three dimensions or any number of them.

What does the theorem actually say?

It is a statement about areas. Draw a right triangle and build a square outward on each of its three sides. The theorem says the big square, on the hypotenuse, has exactly as much area as the two smaller squares put together. Euclid's Elements, written around 300 BCE, states it in just those terms as Book I, Proposition 47: the square on the side opposite the right angle equals the sum of the squares on the sides containing it.

Written as algebra, with a and b for the two shorter sides, called legs, and c for the hypotenuse, that is a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root: c = √(a² + b²). To find a missing leg, subtract instead: a = √(c² − b²). The theorem only holds for right triangles. For any other triangle, a² + b² comes out larger or smaller than c², depending on whether the angle opposite c is less or more than 90 degrees.

How can you see that it is true?

There are hundreds of proofs; one of the oldest and clearest needs no algebra at all, just rearranging pieces. It is a good first example of what a mathematical proof does: show that something must be true for every right triangle, not just the ones you measured.

1. Take four identical copies of a right triangle with legs a and b and hypotenuse c.
2. Draw a large square with sides of length a + b.
3. Arrangement one: put one triangle in each corner of the large square, with its right angle in the corner. The four hypotenuses meet to enclose a tilted square in the middle with side c. The uncovered area is c².
4. Arrangement two: slide the same four triangles into two rectangles, each a by b, pushed into opposite corners of the large square. What is left uncovered now is two squares, one with side a and one with side b. The uncovered area is a² + b².
5. Compare. Both arrangements use the same large square and the same four triangles, so the uncovered area must be the same: c² = a² + b².

The same argument works in symbols. The large square's area is (a + b)², which is a² + 2ab + b². In arrangement one it is also four triangles of area ab/2 each, plus c², so 2ab + c². Setting the two equal and taking 2ab from both sides leaves a² + b² = c².

How is it used in real life?

Squaring a corner. Builders laying out a foundation or a deck need a true right angle, and the theorem run in reverse gives one. Mark 3 feet along one edge from the corner and 4 feet along the other. If the diagonal between the marks measures exactly 5 feet, the corner is square, because 3² + 4² = 5² can only happen with a right angle (Euclid proves that converse as the next proposition, I.48). For a large slab, builders scale up to 6, 8 and 10 feet, or 9, 12 and 15, so that small measuring errors matter less. Sets of whole numbers that fit, like 3, 4, 5 and 5, 12, 13, are called Pythagorean triples.

Straight-line distance. On a city map with a grid, a café is 6 kilometers east and 8 kilometers north of a train station. The straight-line distance is √(6² + 8²) = √(36 + 64) = √100 = 10 kilometers. Walking the streets, east and then north, is 14 kilometers; the theorem gives the distance a drone would fly. Computer games, drawing programs and maps of small areas use this calculation all the time: two points (x₁, y₁) and (x₂, y₂) are √((x₂ − x₁)² + (y₂ − y₁)²) apart.

That map example shows where the flat picture stops being true. Over a city, treating the ground as flat is fine. Over hundreds of kilometers the Earth's curve matters, and distances are worked out along great circles on a sphere, where triangles don't obey a² + b² = c². A triangle drawn on a globe from the North Pole down to the equator, a quarter of the way around, and back up can have three right angles.

Who discovered the Pythagorean theorem?

Not Pythagoras, as far as the evidence shows. A clay tablet known as Plimpton 322, written in Babylonia between about 1800 and 1650 BCE and now at Columbia University, lists the short side and the hypotenuse of right triangles whose sides are all whole numbers, some of them large: its first row gives 119 and 169, which fit a right triangle with sides 119, 120 and 169. Another Old Babylonian tablet, YBC 7289 at Yale, shows a square with its diagonal and an accurate value for the square root of 2, which is the theorem applied to a square (MacTutor History of Mathematics). Indian texts on altar building, the Shulba Sutras, state the rule some time in the first millennium BCE, and Chinese mathematics knows it as the gougu theorem.

Pythagoras lived on the Greek island of Samos and in southern Italy from about 570 to about 495 BCE. None of his writing survives, and no surviving Greek text credits the theorem to him until centuries after his death. The attribution may reflect a real proof by Pythagoras or his followers, or simply his fame. What is certain is that the oldest surviving proof built step by step from stated assumptions is Euclid's, around 300 BCE. Proofs have kept coming since, including one by James Garfield, published in 1876, five years before he became president of the United States.

How does it extend to three dimensions and beyond?

Apply the theorem twice. A box is 3 meters wide, 4 meters deep and 12 meters tall; how long is the longest rod that fits inside, corner to opposite corner? The diagonal across the floor is √(3² + 4²) = 5 meters. That floor diagonal and the height form a new right triangle, so the rod is √(5² + 12²) = √169 = 13 meters. Folding the two steps together gives d = √(x² + y² + z²).

Nothing stops at three. In four, ten or a thousand dimensions, the distance between two points is still the square root of the sum of the squared differences along each direction. That is called Euclidean distance, and it is one of the main ways recommendation systems and machine learning measure how similar two lists of numbers are.

The theorem also leads into trigonometry. Put the hypotenuse at length 1, and the two legs become the sine and cosine of an angle, so the theorem becomes sin² + cos² = 1. Set the hypotenuse to a fixed length r, and the points (x, y) that satisfy x² + y² = r² trace out a circle, the shape behind pi. And a right triangle with both legs 1 has a hypotenuse of √2, which is not a fraction at all, a discovery that forced the Greeks to widen their idea of what a number is.

Check yourself

• Without looking back, explain the rearranged-squares proof in three steps, and say why it works for every right triangle.
• How can a builder with only a tape measure check that a corner is exactly 90 degrees?
• Why doesn't a² + b² = c² hold for a large triangle drawn on the surface of the Earth?

Sources

• Euclid's Elements, Book I, Proposition 47 — David E. Joyce, Clark University
• Euclid's Elements, Book I, Proposition 48 — David E. Joyce, Clark University
• Babylonian Pythagoras — MacTutor History of Mathematics, University of St Andrews
• Plimpton 322 — Wikipedia
• Pythagorean theorem — Wikipedia