Pythagorean Triples: All the Famous Ones and How to Generate Them

A Pythagorean triple is three positive integers (a, b, c) that satisfy a² + b² = c² — the side lengths of a right triangle that happen to be whole numbers. Most right triangles don't have integer sides, but a few special ones do, and these "clean" triangles show up everywhere in geometry tests, construction, and number theory.

The famous triples

(3, 4, 5) — the most basic. Used to lay out 90° angles in carpentry for thousands of years.
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(20, 21, 29)
(9, 40, 41)
(11, 60, 61)
(12, 35, 37)
(13, 84, 85)

Memorize the first three. They appear in disguise on roughly 30% of geometry problems on the SAT, ACT, and standardized state tests.

Primitive vs scaled triples

A primitive triple has no common factor among the three numbers. (3, 4, 5) is primitive. (6, 8, 10) is just (3, 4, 5) doubled — not primitive but still a valid triple. Every multiple of a primitive triple is also a Pythagorean triple:

(3, 4, 5), (6, 8, 10), (9, 12, 15), (12, 16, 20), (300, 400, 500), …
(5, 12, 13), (10, 24, 26), (15, 36, 39), …

This is why a problem might give you a triangle with sides 9, 12, and an unknown — recognize it as a 3-4-5 multiple and the answer (15) drops out instantly.

Euclid's formula: generating all primitive triples

For any two coprime positive integers m > n with one of them even, Euclid's formula generates a primitive Pythagorean triple:

a = m² − n²
b = 2mn
c = m² + n²

Examples:

m = 2, n = 1: a = 3, b = 4, c = 5 ✓
m = 3, n = 2: a = 5, b = 12, c = 13 ✓
m = 4, n = 1: a = 15, b = 8, c = 17 ✓
m = 4, n = 3: a = 7, b = 24, c = 25 ✓
m = 5, n = 2: a = 21, b = 20, c = 29 ✓

This formula generates every primitive Pythagorean triple exactly once. The proof is elegant and was known to Euclid in 300 BC.

Patterns within triples

Several arithmetic patterns make triples easier to spot:

Sum of squares pattern: for any odd a > 1, the triple is (a, (a² − 1)/2, (a² + 1)/2). Try a = 5: (5, 12, 13). a = 7: (7, 24, 25). a = 9: (9, 40, 41).
Even-leg pattern: for any even a > 2, the triple is (a, (a/2)² − 1, (a/2)² + 1). Try a = 6: (6, 8, 10). a = 8: (8, 15, 17).

These let you generate triples mentally without Euclid's formula.

Why triples matter on tests

Standardized tests reuse triples to make geometry problems compute cleanly. If a question gives a right triangle with legs 12 and 16, recognizing 3-4-5 × 4 means the hypotenuse is 20 immediately — no Pythagorean theorem calculation needed. Saving 30 seconds per problem on a test with 50 problems means an extra 25 minutes of clear thinking time.

Memorize at minimum (3,4,5), (5,12,13), (8,15,17), and (7,24,25). All four show up enough that recognizing them earns a noticeable speed advantage.

Pythagorean triples in real life

Carpentry / framing: the "3-4-5 method" for squaring corners. Mark 3 ft along one wall, 4 ft along the perpendicular, and check that the diagonal between marks is exactly 5 ft.
Surveying: Egyptian rope-stretchers used a knotted rope at 3:4:5 spacing to lay out right angles for the pyramids.
Tile and flooring: 3-4-5 triangle templates for cutting precise diagonals.

Number-theoretic depth

Pythagorean triples connect to deep number theory. Fermat's Last Theorem — proved in 1995 by Andrew Wiles — states that no integer solutions exist to aⁿ + bⁿ = cⁿ for any n > 2. The case n = 2 (Pythagorean triples) is special because infinitely many solutions exist.

The proof of Fermat's Last Theorem took 358 years and connects Pythagorean triples to elliptic curves and modular forms — entire branches of modern math.

Verify with the calculator

Our Pythagorean theorem calculator confirms whether any three integers form a triple. Plug in 20, 21, and leave the third blank — the calculator returns 29, confirming (20, 21, 29) is a primitive triple. Useful for verifying patterns or checking unfamiliar triples on harder geometry problems.