Welcome to CalcuGo's Pythagorean Triples Calculator! Don't hesitate to use it whenever you want to quickly and easily check if any given three numbers form a Pythagorean triple. It can also generate Pythagorean triples via the generalized Euclidean formula and verify whether they are primitive!
Not familiar with any of the notions we've used? Don't worry! We'll explain everything in simple terms and provide lots of useful examples. Just scroll down.
🙋 If you've come here to learn about the Pythagorean theorem itself, go check our Pythagorean theorem calculator. For general right-triangle problems, visit the dedicated right triangle calculator.
What is a Pythagorean triple?
The Pythagorean triple definition says it is a set of three positive integers a, b, c that satisfy the relationship:
a² + b² = c²
If you have already learned about the Pythagorean theorem, you surely recognize this formula. Three integers constitute a Pythagorean triple if they are the sides of a right triangle: c is the hypotenuse (the longest side), while a and b are the other two sides, called legs.
To check whether three given integers form a Pythagorean triple, we compute their squares and see if one of them equals the sum of the remaining two squares.
Interestingly, for some triples of integers we can rule them out just by looking at their parity:
- Three odd numbers cannot form a Pythagorean triple. The square of an odd number is odd, so the sum of two odd squares is even — it can never equal a single odd square.
- Two even numbers and one odd number cannot form a Pythagorean triple either.
As a consequence, a Pythagorean triple has to consist of either three even numbers, or two odd numbers and one even number.
Common Pythagorean triples
The best known Pythagorean triple is 3, 4, 5:
- 3² + 4² = 9 + 16 = 25
- 5² = 25
This triple corresponds to the well-known Egyptian triangle. Another famous one is 6, 8, 10 (6² + 8² = 36 + 64 = 100 = 10²). Note that this triple arises from 3, 4, 5 with each number multiplied by 2. Multiplying by 3 gives 9, 12, 15, and so on.
An example that is not related to 3, 4, 5 by multiplication is 5, 12, 13: 5² + 12² = 25 + 144 = 169 = 13².
Generating Pythagorean triples
Given one Pythagorean triple, we can immediately generate infinitely many more by multiplying by positive integers. If a, b, c satisfy a² + b² = c² and n is a positive integer, then n × a, n × b, n × c also satisfy the relationship, because:
(n × a)² + (n × b)² = n²(a² + b²) = n² × c² = (n × c)²
This straightforward construction shows that the set of Pythagorean triples is infinite.
Primitive Pythagorean triples
A Pythagorean triple a, b, c is called primitive if the numbers a, b, c are coprime — that is, their greatest common factor equals one: GCF(a, b, c) = 1.
- 3, 4, 5 is primitive;
- 6, 8, 10 is not primitive because GCF(6, 8, 10) = 2;
- 9, 12, 15 is not primitive because GCF(9, 12, 15) = 3;
- 5, 12, 13 is primitive.
Here is the list of all primitive Pythagorean triples with values less than 100:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)
Primitive triples are the building blocks of all Pythagorean triples — every triple is a primitive one multiplied by a positive integer. If a, b, c is a primitive triple with a² + b² = c², then:
- c is odd;
- exactly one of a and b is odd (the other is even);
- exactly one of a and b is divisible by 3;
- exactly one of a and b is divisible by 4;
- exactly one of a, b, c is divisible by 5.
How to find Pythagorean triples? Pythagorean triples formula
Euclid's formula lets us generate Pythagorean triples from pairs of positive integers. Let m and n be positive integers with m > n. Then the three numbers
- a = m² − n²
- b = 2 × m × n
- c = m² + n²
form a Pythagorean triple. This triple is primitive if and only if GCF(m, n) = 1 and exactly one of m, n is odd. Every primitive Pythagorean triple arises from Euclid's formula from a unique pair of integers m, n — so there are infinitely many primitive triples.
Euclid's basic formula cannot produce every non-primitive triple (for instance, it cannot directly generate 9, 12, 15). To reach all triples, we add a positive integer multiplier k:
- a = (m² − n²) × k
- b = 2 × m × n × k
- c = (m² + n²) × k
This calculator uses exactly this generalization of Euclid's formula to generate Pythagorean triples.
How to use this Pythagorean triple calculator
- Pick a mode: "Check a triple" to test three numbers, or "Generate a triple" to build one from m, n, and k.
- Select your measurement system — Metric (mm, cm, m, km) or Imperial/US (in, ft, yd, mi) — used when the sides are read as lengths for area and perimeter.
- Check mode: enter three positive integers a, b, c. The tool reports whether they form a Pythagorean triple, which value is the hypotenuse, whether the triple is primitive, and its greatest common divisor.
- Generate mode: enter integers m > n > 0 and a multiplier k ≥ 1. The tool builds the triple (a, b, c), tells you whether it is primitive, and shows the full step-by-step formula and verification.
Pythagoras days
A Pythagoras day is a calendar date whose day, month, and year form a Pythagorean triple. For example, take the date 16/12/20 (16 December 2020): 16² + 12² = 256 + 144 = 400 = 20². Another one is 8/6/10 (8 June 2010), since 6² + 8² = 10². These playful dates are rare and mathematically pleasing — a fun way to spot Pythagorean triples in everyday life.
Frequently Asked Questions
What is the smallest Pythagorean triple?
The smallest and most famous Pythagorean triple is 3, 4, 5. It is also primitive, meaning its three numbers share no common factor greater than 1.
Is every multiple of a Pythagorean triple also a triple?
Yes. If (a, b, c) is a Pythagorean triple, then (n·a, n·b, n·c) is also a triple for any positive integer n. Only the original — when the numbers are coprime — is primitive.
How many Pythagorean triples are there?
Infinitely many. Since every triple can be scaled by any positive integer, and there are infinitely many primitive triples via Euclid's formula, the total count is unbounded.
Can Pythagorean triples contain non-integers?
No — by definition a Pythagorean triple consists of three positive integers. Three non-integer side lengths can still satisfy a² + b² = c² and form a right triangle, but they are not called a Pythagorean triple.