Pythagorean Triples
Whole-number sets that satisfy a² + b² = c².
About this topic
A Pythagorean triple is a set of three whole numbers satisfying a² + b² = c², such as (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25). Multiplying a triple by any whole number gives another, like (6, 8, 10).
Recognising them saves calculation and helps with quick right-angle checks, for example builders using a 3-4-5 rope to make a square corner. Euclid's formula (m² − n², 2mn, m² + n²) generates them all.
Worked example: a builder wants a right angle for a patio corner. Marking 3 m along one wall and 4 m along another, she adjusts until the diagonal between marks is exactly 5 m. Scaling up, 30, 40 and 50 also works. To generate a triple, take m = 3 and n = 2: sides 9 − 4 = 5, 2·3·2 = 12 and 9 + 4 = 13, giving (5, 12, 13).
Ancient Egyptians and Babylonians knew these triples, and the Plimpton 322 clay tablet lists many. Students sometimes think every right triangle has whole number sides, but most, like legs 1 and 1 with hypotenuse √2, do not. Also be careful that a triple needs a² + b² = c² exactly, so (4, 5, 6) fails since 16 + 25 = 41, not 36. Memorising the first few triples lets you spot answers quickly on tests without a calculator.
Key ideas
- 3-4-5 and 5-12-13 are common
- Multiples also work
- Great for quick checks
Quick check
Use cases: where Pythagorean Triples helps
- Homework: review the key idea behind Pythagorean Triples before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Pythagorean Triples without notes.
- Visual learning: play with the interactive lab to see how Pythagorean Triples changes when you adjust its inputs.
- Teaching a friend: explain Pythagorean Triples in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Pythagorean Triples?
Whole-number sets that satisfy a² + b² = c².
What will I learn in Pythagorean Triples?
3-4-5 and 5-12-13 are common. Multiples also work. Great for quick checks.
Is Pythagorean Triples part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Pythagorean Triples lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Pythagorean Triples?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.