Mathematics

Pythagorean Triples

Three whole numbers with a² + b² = c². The four triples worth memorising, why their multiples count too, and the check that spots one in a timed exam.

Pythagorean triple

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.

Also called: Pythagorean triad
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.

The short 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.

An example

8² + 15² = 64 + 225 = 289 = 17²

The two shorter sides squared and added give exactly the square of the longest side, so 8, 15 and 17 is a triple and a triangle with those sides is right-angled. The same test works in reverse as a proof: if a question gives you a triangle with sides 8 cm, 15 cm and 17 cm and asks whether the angle between the two shorter sides is 90°, this line is the whole answer.

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.

Common 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.

Last updated August 21, 2026