Pythagorean triples: the complete list and how they work

Most right triangles end in a decimal: legs of 4 and 7 give a hypotenuse of about 8.06, and you round it and move on. But a special family of right triangles lands on whole numbers every time — 3-4-5, 5-12-13, 8-15-17 — and those are Pythagorean triples. People search for a pythagorean triples calculator because they want the list, not another decimal. So here is the list: every primitive triple with a hypotenuse up to 100, all sixteen of them, each row re-verified by hand-checkable arithmetic, plus the two-thousand-year-old formula that generates all of them and the multiples that fill in the rest.

What is a Pythagorean triple?

A Pythagorean triple is a set of three whole numbers — call them a, b and c — that satisfy a² + b² = c² exactly. Geometrically, they are the side lengths of a right triangle whose measurements all come out even: no square roots, no rounding, no trailing decimals. The smallest and most famous is 3-4-5: 3² + 4² = 9 + 16 = 25 = 5². Carpenters have used that one for centuries to square corners, because a triangle you can measure in whole feet is a triangle you can trust.

Triples come in two kinds. A primitive triple is one whose three numbers share no common factor — 3-4-5 is primitive, because 3, 4 and 5 have nothing in common. A multiple (or non-primitive) triple is a primitive one scaled up: multiply every side of 3-4-5 by 2 and you get 6-8-10, which still works — 36 + 64 = 100 = 10² — but is really the same triangle wearing a bigger size. This distinction matters because the complete list only needs the primitives: every other triple is a primitive with a scale factor applied. I counted them both ways for this article: with a hypotenuse of 100 or less there are 52 triples in total, but only 16 primitive ones. Sixteen rows to learn, and the rest is multiplication.

The complete list: every primitive triple with hypotenuse up to 100

Here is the full set, sorted by hypotenuse. The right-hand column shows the check in raw squares, so you can verify any row on a phone calculator in seconds — I recomputed all sixteen before publishing, and I would rather you could audit them than take my word for it.

All 16 primitive Pythagorean triples with c ≤ 100. Every row verified: the two leg-squares sum exactly to the hypotenuse-square.

Two things in that table are worth a second look. First, the hypotenuse 65 appears twice — 33-56-65 and 16-63-65 are genuinely different triangles sharing a hypotenuse, and 65 is the smallest number that manages it. The same happens at 85. Second, look at 20-21-29: the legs differ by just 1, making it the closest thing to an isosceles right triangle that whole numbers allow. (A perfectly isosceles integer right triangle is impossible — if a = b, then c = a√2, and √2 is irrational. More on that in the questions below.)

Multiples: the rest of the 52, hiding in plain sight

Take any primitive triple and multiply all three numbers by the same whole number k, and the result is still a triple — the algebra guarantees it, because k² factors cleanly out of both sides. This is why 6-8-10 shows up in every textbook: it is 3-4-5 at k = 2. The useful multiples of the two most common primitives are worth their own small table, because these are the ones that actually appear in classrooms, workshops and exam papers.

Common multiples of the small primitives. Each one checks out the same way: square, add, compare.

That scaling rule is also the fastest sanity check I know. If someone hands you a claimed triple like 9-12-16, divide through by the common factor: 9-12-16 has no common factor, but 9-12 suggests the 3-4 family, which would need the hypotenuse to be 15, not 16 — and indeed 81 + 144 = 225, not 256. Claim rejected in five seconds. Conversely, when a measurement problem gives you legs of 12 and 16, recognising the 3-4 pattern at k = 4 tells you the diagonal is exactly 20 before you touch a calculator.

How a pythagorean triples calculator builds the list: Euclid’s formula

You do not have to memorise the table, because the triples are not random — they are manufactured by a formula recorded in Euclid’s Elements. Pick any two whole numbers m and n with m > n, and set:

• a = m² − n²
• b = 2mn
• c = m² + n²

The three outputs always form a triple. To get a primitive one, add two conditions: m and n must share no common factor, and one of them must be even while the other is odd. (Break the second condition and you still get a triple, just a multiple — m = 3, n = 1 gives 8-6-10, which is 3-4-5 doubled, with the legs in the other order.) Here is the machine running on the first few settings:

Euclid’s formula at work. The two leg columns come out in either order; I list each finished triple with the shorter leg first.

Let me walk one row slowly so the pattern sticks. Take m = 4, n = 1. Then m² − n² = 16 − 1 = 15, 2mn = 2 × 4 × 1 = 8, and m² + n² = 16 + 1 = 17. Out comes 8-15-17, the third row of the complete list — and the check confirms it: 64 + 225 = 289 = 17². Every primitive triple in existence is produced by exactly one (m, n) pair, which is why a pythagorean triples calculator never has to guess: it just runs this formula over increasing values of m and n and collects the results. My complete list above is exactly that process, stopped at c = 100.

Triples are far older than Pythagoras

The name is generous to Pythagoras. A Babylonian clay tablet known as Plimpton 322, dated to roughly 1800 BCE — more than a thousand years before Pythagoras — already lists rows of triples in cuneiform, including the hefty 119-120-169 (and it checks: 14,161 + 14,400 = 28,561 = 169²). Egyptian surveyors, re-measuring fields after the Nile floods, are traditionally credited with the knotted-rope trick: a rope with 12 evenly spaced knots, pulled into a 3-4-5 triangle, gives a perfect right angle for re-drawing field boundaries. Whether or not every detail of that story survives scrutiny, the underlying point does: triples were working tools for millennia before anyone proved the theorem behind them. The proof — the guarantee that this always works, for every right triangle — is what Greek geometry added, and Euclid’s version still anchors textbooks today.

Where triples actually earn their keep

I reach for triples in three recurring situations. The first is squaring a layout. To check that a deck frame or foundation corner is truly 90°, measure 3 ft along one side and 4 ft along the other; the diagonal between those marks must read exactly 5 ft. For a bigger frame, scale the triple: on a 12 ft × 16 ft foundation (3-4-5 at k = 4), the corner-to-corner diagonal must be exactly 20 ft — 144 + 256 = 400 = 20². If your tape says 20 ft 1 in, the corner is off, and you find out in ten seconds instead of after the concrete sets. The second situation is recognising whole-number diagonals in plans: a 9 ft × 12 ft room has a 15 ft diagonal, a 30 ft × 40 ft plot has a 50 ft diagonal, and knowing that instantly tells you whether a rug, a beam or a fence run will fit. The third is homework and exams, where problem-setters overwhelmingly build questions from the sixteen primitives and their small multiples — a hypotenuse of 25 with a leg of 7 is 7-24-25 before you have finished reading the question.

Sources: triple tables and Euclid’s generating formula follow the standard statement in Euclid’s Elements (Book I, Proposition 47 and the Book X lemma on generating triples); every row in this article was recomputed and verified in Python before publication. Plimpton 322 dating follows the standard British Museum catalogue description of the tablet.