5-12-13 Triangle: Properties, Area, Angles & Pythagorean Proof

What is the 5-12-13 triangle?

The 5-12-13 triangle is a right-angled triangle whose three sides are exactly 5, 12, and 13 units long. It is one of the best-known Pythagorean triples — a set of three whole numbers that satisfy the Pythagorean theorem. Because all three sides are whole numbers, it appears constantly in geometry classes, exam questions, and real-world building and design.

Why it is a right triangle (the Pythagorean proof)

A triangle is right-angled when the square of its longest side equals the sum of the squares of the other two. For the 5-12-13 triangle: 5² + 12² = 25 + 144 = 169, and 13² = 169. Since the two sides match, the theorem holds exactly, so the angle opposite the side of length 13 is a perfect 90°. The longest side, 13, is the hypotenuse.

Area, perimeter and angles

Because the two shorter sides (5 and 12) meet at the right angle, they act as the base and height, so the area is ½ × 5 × 12 = 30 square units. The perimeter is simply 5 + 12 + 13 = 30 units. The three angles are 90° plus two acute angles: about 22.62° (opposite the side of 5) and 67.38° (opposite the side of 12), which you can confirm with our trigonometry calculator.

Is 5-12-13 a primitive Pythagorean triple?

Yes. A Pythagorean triple is “primitive” when its three numbers share no common factor other than 1. Since 5, 12, and 13 have no common divisor, the set is primitive. Multiplying every side by the same number (for example 10-24-26) creates a larger but similar right triangle that is no longer primitive.

Where the 5-12-13 triangle is used

Builders and carpenters use Pythagorean triples like 3-4-5 and 5-12-13 to check that corners are perfectly square without a set square — measure the sides, and if they match the triple, the angle is exactly 90°. The triangle also appears throughout trigonometry, navigation, and engineering. To explore any triangle’s measurements, try our trigonometry calculator or the scientific calculator.

Frequently asked questions

What is the hypotenuse of a 5-12-13 triangle?

The hypotenuse is 13 — the longest side, opposite the right angle.

What is the area of a 5-12-13 triangle?

30 square units, found with ½ × base × height = ½ × 5 × 12.