Pythagorean Theorem Calculator

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pythagorean triples

Pythagorean Triples

A Pythagorean triple (a, b, c) satisfies a²+b²=c² with all three values positive integers. Euclid's formula generates any primitive triple from two numbers m and n: a=m²−n², b=2mn, c=m²+n², with m>n>0, coprime, and m−n odd.

Generating, scaling and checking Pythagorean triples

With m=4 and n=3 (coprime, odd difference), Euclid's formula gives a=16−9=7, b=2×4×3=24 and c=16+9=25, forming the triple (7,24,25). That triple can be checked with the site's own hypotenuse calculator: entering legs 7 and 24 returns c=√(49+576)=√625=25 exactly, with no decimal places, confirming the three numbers satisfy a²+b²=c².
Not every triple comes directly from Euclid's formula: multiplying all three sides of a primitive triple by the same integer always produces another valid triple, just not a primitive one. The (3,4,5) triple multiplied by 4 gives (12,16,20), and the same hypotenuse calculator confirms it: legs 12 and 16 return c=√(144+256)=√400=20. A triple is called primitive when the greatest common divisor of a, b and c is 1; (12,16,20) is not primitive because all three sides share the factor 4.
There are exactly 16 primitive triples with a hypotenuse under 100, found by summing every valid (m,n) pair up to that limit. A curious case: 65 is the hypotenuse of two different primitive triples at once, (16,63,65) and (33,56,65), because 65 can be written as the sum of two squares in two ways: 65=1²+8² and 65=4²+7². Each of those sums generates a different (m,n) pair in Euclid's formula, and each pair produces a distinct primitive triple sharing the same hypotenuse.

Examples

Generating (7,24,25) with m=4, n=3

Input: m=4, n=3
Expected output: a=7, b=24, c=25

7²+24²=625=25²; the hypotenuse calculator confirms c=25 from legs 7 and 24.

Scaling (3,4,5) by 4

Input: a=12, b=16, c=20
Expected output: 144+256=400 ✓

Not a primitive triple: all three sides share the factor 4 with the original (3,4,5) triple.

Two primitive triples with hypotenuse 65

Input: c=65
Expected output: (16,63,65) e (33,56,65)

65=1²+8²=4²+7², and each sum of squares generates a different (m,n) pair in Euclid's formula.

Full tool FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Attributed to Greek mathematician Pythagoras (~570–495 BC).

What is the hypotenuse?

What are the legs?

What is a Pythagorean triple?

How can I check whether a triangle is a right triangle?

Does the tool work with decimal values?

How precise is the result?

Does the theorem apply to non-right triangles?

Frequently asked questions

How do I quickly check if three numbers form a Pythagorean triple?

Enter the two smaller numbers as legs in the site's hypotenuse calculator: if the returned hypotenuse matches the third number exactly, with no decimal places, the three form a triple. For 7 and 24, for example, the result is exactly 25.

Does every Pythagorean triple come from Euclid's formula?

Every primitive triple comes from some coprime (m,n) pair with an odd difference. Non-primitive triples, however, are just integer multiples of a primitive one, like (12,16,20), which is (3,4,5) multiplied by 4, and do not correspond to their own (m,n) pair in the formula.

Can a hypotenuse belong to more than one primitive triple?

Yes. The best-known case is 65, the hypotenuse of both (16,63,65) and (33,56,65), because 65 can be written as the sum of two squares in two different ways (1²+8² and 4²+7²), and each way feeds a distinct (m,n) pair into Euclid's formula.

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pythagorean triples

Pythagorean Triples

Generating, scaling and checking Pythagorean triples

With m=4 and n=3 (coprime, odd difference), Euclid's formula gives a=16−9=7, b=2×4×3=24 and c=16+9=25, forming the triple (7,24,25). That triple can be checked with the site's own hypotenuse calculator: entering legs 7 and 24 returns c=√(49+576)=√625=25 exactly, with no decimal places, confirming the three numbers satisfy a²+b²=c².

Not every triple comes directly from Euclid's formula: multiplying all three sides of a primitive triple by the same integer always produces another valid triple, just not a primitive one. The (3,4,5) triple multiplied by 4 gives (12,16,20), and the same hypotenuse calculator confirms it: legs 12 and 16 return c=√(144+256)=√400=20. A triple is called primitive when the greatest common divisor of a, b and c is 1; (12,16,20) is not primitive because all three sides share the factor 4.

There are exactly 16 primitive triples with a hypotenuse under 100, found by summing every valid (m,n) pair up to that limit. A curious case: 65 is the hypotenuse of two different primitive triples at once, (16,63,65) and (33,56,65), because 65 can be written as the sum of two squares in two ways: 65=1²+8² and 65=4²+7². Each of those sums generates a different (m,n) pair in Euclid's formula, and each pair produces a distinct primitive triple sharing the same hypotenuse.

Examples

Generating (7,24,25) with m=4, n=3

Input: m=4, n=3
Expected output: a=7, b=24, c=25

7²+24²=625=25²; the hypotenuse calculator confirms c=25 from legs 7 and 24.

Scaling (3,4,5) by 4

Input: a=12, b=16, c=20
Expected output: 144+256=400 ✓

Not a primitive triple: all three sides share the factor 4 with the original (3,4,5) triple.

Two primitive triples with hypotenuse 65

Input: c=65
Expected output: (16,63,65) e (33,56,65)

65=1²+8²=4²+7², and each sum of squares generates a different (m,n) pair in Euclid's formula.

Full tool FAQ

What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Attributed to Greek mathematician Pythagoras (~570–495 BC).

What is the hypotenuse?

What are the legs?

What is a Pythagorean triple?

How can I check whether a triangle is a right triangle?

Does the tool work with decimal values?

How precise is the result?

Does the theorem apply to non-right triangles?

Frequently asked questions