Pythagorean Theorem: a² + b² = c² Explained Visually

The right-angled triangle

A right triangle has one corner of exactly 90°, like the corner of a sheet of paper. The two sides that make the right angle are the legs, called a and b. The long side opposite it is the hypotenuse, c, and it is always the longest side. A popular story says that rope-stretchers in ancient Egypt made a right angle with a rope tied in twelve equal steps by knots, bent into a triangle with sides of 3, 4 and 5 steps. The numbers fit, since 3 + 4 + 5 = 12. But no Egyptian text describes this, so treat it as a legend rather than a fact.

Squares on the sides

Draw a square on each side of the 3-4-5 triangle. The square on the short leg has area 3 × 3 = 9, and the one on the other leg has 4 × 4 = 16. The square on the hypotenuse has 5 × 5 = 25. And 9 + 16 = 25. This is no accident: it works for every right triangle. In words, the two small squares together have exactly the area of the big square. In symbols, a² + b² = c², where a and b are the legs and c is the hypotenuse. It holds only when the angle is a right angle.

Finding a missing side

The rule lets you find a side you cannot measure. Take legs a = 5 and b = 12. Then c² = 25 + 144 = 169, and c = √169 = 13. Drag the sliders in the sandbox and the squares grow to scale beside the triangle. Many choices give whole numbers, like 6-8-10 or 8-15-17, but most do not: with legs 1 and 1, c = √2 ≈ 1.414. To find a leg instead, subtract. If c = 10 and a = 6, then b² = 100 − 36 = 64 and b = 8. Whenever you can, check that c comes out as the longest side.

A proof without algebra

Here is one of the oldest ideas for proving it. Take four identical right triangles and place them inside a square with side a + b. Arranged one way, they leave a tilted square of side c in the middle. Arranged another way, they leave two squares, one of side a and one of side b. The big square and the four triangles never change, so the empty area must be the same both times: c² = a² + b². Check it with 3 and 4. The big square has area 7 × 7 = 49, four triangles take 4 × 6 = 24, and 49 − 24 = 25 is left, which is 9 + 16.

Older than Pythagoras

Pythagoras lived around 570 to 495 BC, on the Greek island of Samos and later in southern Italy, and he left no writings of his own, so his part in the theorem is unclear. Much older is Plimpton 322, a clay tablet from Babylon made about 1800 BC and now kept at Columbia University in New York. Its rows hold numbers that belong to right triangles, such as 119, 120 and 169, and indeed 119² + 120² = 14 161 + 14 400 = 28 561 = 169². Scholars still argue about what the tablet was for. Texts from India and China also use the rule.

Distance on a map

Say you walk 3 km east and then 4 km north. How far are you from the start in a straight line? The two walks make the legs of a right triangle, so the distance is √(3² + 4²) = √25 = 5 km. Maps and games use this all the time: the distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). Televisions work the same way. A TV that is 48 inches wide and 27 inches tall has a diagonal of √(48² + 27²) = √3033 ≈ 55 inches, which is why it is sold as a 55-inch set.

Into three dimensions

Use the theorem twice and you can cross a box. Take a box 3 wide, 4 deep and 12 high. The diagonal across its floor is √(3² + 4²) = 5. Now take that line and the height 12 as the legs of a new right triangle: the long diagonal through the middle of the box is √(5² + 12²) = √169 = 13. In one step, d = √(a² + b² + c²) for edges a, b and c, and indeed √(9 + 16 + 144) = 13. This tells you the longest rod that fits inside a box. In the drawing the wireframe box turns slowly and its diagonal lights up.

Pythagoras and right triangles