5-12-13 triangle — Definition, Formula & Examples
A 5-12-13 triangle is a right triangle whose three sides are in the exact ratio 5 : 12 : 13. Because all three side lengths are whole numbers that satisfy the Pythagorean theorem, 5-12-13 is a Pythagorean triple.
The ordered triple is a primitive Pythagorean triple, meaning with . Any right triangle with sides , , and for positive real is called a 5-12-13 triangle.
Key Formula
Where:
- = One leg of the right triangle
- = The other leg of the right triangle
- = The hypotenuse (longest side)
How It Works
Whenever you spot sides in a 5 : 12 : 13 ratio, you immediately know the triangle is a right triangle — no need to compute angles or run the full Pythagorean theorem. The legs are 5 and 12, and the hypotenuse (the longest side, opposite the right angle) is 13. You can scale the triple by any factor : for instance, -- (with ) or -- (with ) are also right triangles. This shortcut saves time on standardized tests and in construction problems where you need to verify a right angle quickly.
Worked Example
Problem: A right triangle has legs of length 15 cm and 36 cm. Find the hypotenuse without a calculator.
Recognize the ratio: Divide each leg by a common factor. 15 ÷ 3 = 5 and 36 ÷ 3 = 12. The legs are in the ratio 5 : 12, so this is a scaled 5-12-13 triangle with k = 3.
Find the hypotenuse: Multiply 13 by the same scale factor.
Verify: Check with the Pythagorean theorem.
Answer: The hypotenuse is 39 cm.
Why It Matters
Memorizing a few Pythagorean triples like 5-12-13 lets you solve right-triangle problems on the SAT, ACT, and geometry exams in seconds instead of minutes. It also appears in real-world layout tasks — carpenters and surveyors use integer triples to check that corners form true 90° angles.
Common Mistakes
Mistake: Confusing which side is the hypotenuse and placing 13 as a leg instead of the longest side.
Correction: The hypotenuse is always the largest number in a Pythagorean triple and sits opposite the right angle. In 5-12-13, the hypotenuse is 13.