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Pythagorean Triangle — Definition, Formula & Examples

A Pythagorean triangle is a right triangle whose three side lengths are all positive integers, meaning they form a Pythagorean triple. For example, a triangle with sides 3, 4, and 5 is a Pythagorean triangle because all sides are whole numbers and 32+42=523^2 + 4^2 = 5^2.

A Pythagorean triangle is a right triangle with side lengths (a,b,c)(a, b, c) where a,b,c∈Z+a, b, c \in \mathbb{Z}^+ and a2+b2=c2a^2 + b^2 = c^2, with cc being the hypotenuse. The ordered triple (a,b,c)(a, b, c) is called a Pythagorean triple. A primitive Pythagorean triangle is one where gcd⁡(a,b,c)=1\gcd(a, b, c) = 1.

Key Formula

a2+b2=c2a^2 + b^2 = c^2
Where:
  • aa = One leg of the right triangle (positive integer)
  • bb = The other leg of the right triangle (positive integer)
  • cc = The hypotenuse — the longest side (positive integer)

How It Works

To check whether a right triangle is a Pythagorean triangle, verify that all three side lengths are positive integers and that they satisfy the Pythagorean theorem. Start with the two shorter sides aa and bb, compute a2+b2a^2 + b^2, and check whether the result is a perfect square. If it is, and the square root is a whole number equal to the hypotenuse cc, you have a Pythagorean triangle. You can also generate new Pythagorean triangles by multiplying every side of a known triple by the same positive integer — for instance, scaling (3,4,5)(3, 4, 5) by 2 gives (6,8,10)(6, 8, 10).

Worked Example

Problem: Determine whether a right triangle with sides 5, 12, and 13 is a Pythagorean triangle.
Step 1: Identify the two legs and the hypotenuse. The longest side, 13, is the hypotenuse.
a=5,b=12,c=13a = 5,\quad b = 12,\quad c = 13
Step 2: Check that all three sides are positive integers. Yes — 5, 12, and 13 are all whole numbers.
Step 3: Verify the Pythagorean theorem by computing the sum of the squares of the legs.
52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169
Step 4: Compare to the square of the hypotenuse.
132=16913^2 = 169
Answer: Since 52+122=1325^2 + 12^2 = 13^2 and all sides are positive integers, this is a Pythagorean triangle. The triple (5, 12, 13) is also primitive because the three numbers share no common factor greater than 1.

Another Example

Problem: Is a right triangle with legs 9 and 12 a Pythagorean triangle?
Step 1: Compute the sum of the squares of the two legs.
92+122=81+144=2259^2 + 12^2 = 81 + 144 = 225
Step 2: Take the square root to find the hypotenuse.
225=15\sqrt{225} = 15
Step 3: Since 15 is a whole number, the sides (9, 12, 15) form a Pythagorean triple. Notice that this triple equals 3×(3,4,5)3 \times (3, 4, 5), so it is not primitive.
Answer: Yes, the triangle with sides 9, 12, and 15 is a Pythagorean triangle. It is a scaled version of the (3, 4, 5) triple.

Visualization

Why It Matters

Pythagorean triangles appear throughout middle-school and high-school geometry whenever you need clean integer answers for distance, area, or diagonal problems. Builders and carpenters use the 3-4-5 triple to verify that corners are square. In number theory courses, studying which integers form Pythagorean triples leads to deeper results, including the proof of Fermat's Last Theorem.

Common Mistakes

Mistake: Assuming every right triangle is a Pythagorean triangle.
Correction: A right triangle with legs 1 and 1 has hypotenuse 2\sqrt{2}, which is not an integer. A Pythagorean triangle specifically requires all three sides to be whole numbers.
Mistake: Forgetting that scaled triples still count as Pythagorean triangles.
Correction: The triple (6, 8, 10) is a valid Pythagorean triple even though it equals 2×(3,4,5)2 \times (3, 4, 5). It just is not a primitive triple.

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