The Theorem
In any right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:
a² + b² = c²
Rearranged, it finds any missing side: c = √(a² + b²), a = √(c² − b²). The hypotenuse c is always the longest side and always sits opposite the right angle.
Pythagorean Triple Chart
Whole-number sides that satisfy the theorem exactly:
| a | b | c | Notes |
|---|---|---|---|
| 3 | 4 | 5 | The smallest; used to square corners on site |
| 5 | 12 | 13 | Common in textbooks |
| 8 | 15 | 17 | — |
| 7 | 24 | 25 | — |
| 20 | 21 | 29 | Nearly isosceles |
| 9 | 40 | 41 | — |
| 12 | 35 | 37 | — |
Any multiple of a triple is also a triple: 6-8-10 and 9-12-15 are both scaled versions of 3-4-5. There are infinitely many primitive triples, generated by Euclid's formula from any two coprime integers.
The 3-4-5 Method
Builders use the theorem in reverse to check that a corner is square. Measure 3 units along one wall, 4 along the other, and the diagonal between the marks must be exactly 5. If it is not, the corner is not a right angle. Larger multiples — 6-8-10, 12-16-20 — give better accuracy over longer walls.
Older Than Pythagoras
The Babylonian tablet Plimpton 322, dating from around 1800 BC, lists Pythagorean triples more than a millennium before Pythagoras. Egyptian rope-stretchers and Chinese and Indian mathematicians all used the relationship independently. Pythagoras or his school are credited with an early proof rather than the discovery, and the theorem now has several hundred distinct proofs, including one by US President James Garfield.
Where It Is Used
- The distance formula — the theorem applied to coordinates.
- Screen and TV sizes — a 55-inch diagonal on a 16:9 panel is 47.9 by 26.9 inches.
- Navigation and surveying — converting between components and straight-line distances.
- Vector magnitudes — the length of a vector is the Pythagorean sum of its components.
- Standard deviation — the same square-and-root structure appears throughout statistics.
Frequently Asked Questions
Does it work for non-right triangles?
No. The generalisation is the law of cosines: c² = a² + b² − 2ab cos(C). When C is 90°, the cosine term vanishes and it reduces to the Pythagorean theorem.
How do I know which side is the hypotenuse?
It is opposite the right angle and always the longest side. If a calculation makes a leg longer than the hypotenuse, the labels have been swapped.
Can the sides be decimals?
Yes. Whole-number solutions are special cases; the theorem holds for any real lengths.