---
title: "Pythagorean Triples"
url: https://www.learningloftinstitute.com/glossary/pythagorean-triple
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › Pythagorean Triples"
---
# Pythagorean Triples
> **Answer.** A Pythagorean triple is three whole numbers satisfying a² + b² = c², such as 3, 4 and 5. A triangle with those side lengths has a right angle. Four triples are worth memorising — 3-4-5, 5-12-13, 8-15-17 and 7-24-25 — because exam questions are built out of them and their multiples.
## Key facts
| | |
| --- | --- |
| Definition | A Pythagorean triple is a set of three positive whole numbers a, b and c for which a² + b² = c², so they form a right-angled triangle. |
| Where students meet it | Grade 8, as soon as Pythagoras' theorem is introduced, and repeatedly afterwards in IGCSE 0580 and Digital SAT geometry, where side lengths are chosen to come out whole. |
| Also known as | Pythagorean triad |
## The four worth memorising
There are infinitely many triples, but exam papers reuse a short list because they want clean numbers. Learning four of them means that a good proportion of Pythagoras and trigonometry questions can be finished by recognition, with the calculation done only as a check.
These four are primitive, meaning the three numbers share no common factor. Every other useful triple is one of these scaled up, or one of a handful of rarer primitives such as 9-40-41 and 20-21-29.
- 3-4-5 — because 9 + 16 = 25
- 5-12-13 — because 25 + 144 = 169
- 8-15-17 — because 64 + 225 = 289
- 7-24-25 — because 49 + 576 = 625
## Every multiple is a triple too
Multiply all three numbers of a triple by the same whole number and the result is still a triple, because both sides of a² + b² = c² are multiplied by the square of that factor. So 6-8-10, 9-12-15 and 30-40-50 are all the 3-4-5 triangle at different sizes, and the triangles are similar — same angles, different scale.
This is why a question about a 10 cm and 24 cm right-angled triangle is really a 5-12-13 question doubled, giving 26 cm without touching a calculator.
- 3-4-5 → 6-8-10, 9-12-15, 12-16-20, 15-20-25, 30-40-50
- 5-12-13 → 10-24-26, 15-36-39
- 8-15-17 → 16-30-34
- 7-24-25 → 14-48-50
## Spotting one under time pressure
When two sides are given, divide them by their highest common factor and see whether a familiar pair appears. Sides of 21 and 28 have an HCF of 7 and reduce to 3 and 4, so the hypotenuse is 7 × 5 = 35. Sides of 45 and 108 reduce to 5 and 12, giving 45 × 13 ÷ 5, which is 117.
Two honest limits on this. It is a shortcut for arithmetic, not a substitute for the theorem — most right-angled triangles have irrational sides and no triple in sight, and a question whose answer is 5√2 is not broken. And recognising a triple only tells you which side is the hypotenuse if you know that the two numbers you were given are the two shorter sides; if one of them is the hypotenuse, you are subtracting, not adding.
## Questions
### Is 6-8-10 a Pythagorean triple?
Yes. It satisfies 36 + 64 = 100, so it qualifies. It is not a primitive triple, because all three numbers divide by 2 to give 3-4-5, but nothing in the definition requires the numbers to be coprime. The triangle it describes is a 3-4-5 triangle enlarged by scale factor 2.
### How can I tell whether a triangle is right-angled?
Square the two shorter sides, add them, and compare with the square of the longest side. Equal means right-angled. If the sum is larger, the angle opposite the longest side is acute; if it is smaller, that angle is obtuse. Exam questions phrase this as "show that triangle ABC is right-angled", and the two lines of squaring are the whole proof.
### Are there triples other than these four?
Infinitely many. Every pair of whole numbers m > n generates one through a = m² − n², b = 2mn, c = m² + n². Taking m = 2, n = 1 produces 3-4-5; m = 3, n = 2 produces 5-12-13. School papers stay with the small ones, so the four listed here plus their multiples cover almost everything you will meet.
### Do I still need to memorise triples if I have a calculator?
You do not need to, but it changes how a paper feels. Recognising 5-12-13 turns a two-minute calculation into a glance, and the time saved goes to the questions that actually need thinking. It also gives you an instant check on a calculator answer, which matters most on a non-calculator paper where the arithmetic is yours to do.
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