The 5-12-13 Triangle

A 5-12-13 triangle is a right triangle with legs 5 and 12 and hypotenuse 13. It is a right triangle because 5² + 12² = 25 + 144 = 169 = 13². Its acute angles are about 22.62° and 67.38°, and after 3-4-5 it is the smallest primitive Pythagorean triple: 5, 12 and 13 share no common factor.

The legs, 5 and 12, meet at the right angle; the hypotenuse, 13, is the longest side, opposite the right angle. Because the numbers are small and the angles are not "nice", the 5-12-13 triangle is a favorite of test writers: you can't guess your way through it, but every answer comes out exact.

Why 5, 12 and 13 form a right triangle

The quickest check is the converse of the Pythagorean theorem: if the squares of the two shorter sides add up to the square of the longest, the triangle has a right angle. Here 25 + 144 = 169, so it does.

There is also a neat picture behind those numbers. A 13-by-13 square is a 12-by-12 square plus a one-unit-wide L-shaped border. That border holds 13 + 12 = 25 unit squares, which is exactly enough to build a 5-by-5 square.

In algebra, that picture is the difference of two squares: 13² − 12² = (13 − 12)(13 + 12) = 1 × 25 = 25. Whenever the hypotenuse is exactly one more than a leg, the leftover border is a single strip, and the triple works when that strip contains a perfect square number of cells. That idea generates a whole family of triples, explained further down.

5-12-13 triangle angles

One angle is 90°. Call the angle opposite the side of length 5 angle A, and the angle opposite the side of length 12 angle B. Using the tangent ratio (opposite over adjacent):

Both angles are rounded to the nearest hundredth of a degree; they add to 90°, as the two acute angles of every right triangle must. The sine and cosine ratios are exact fractions:

These exact ratios are why the triangle appears in so many trigonometry questions. A typical one: "sinθ = 5 over 13 and θ is acute; find cosθ and tanθ." Draw a right triangle with opposite side 5 and hypotenuse 13, recognize the triple, and the adjacent side is 12, so cosθ = 12 over 13 and tanθ = 5 over 12. The Pythagorean identity sin²θ + cos²θ = 1 gives the same answer: 25 over 169 + 144 over 169 = 1.

Properties of the 5-12-13 triangle

Decimals are rounded to the nearest hundredth. Two entries deserve a second look. First, the area and the perimeter are the same number, 30. Only one other right triangle with whole-number sides can say that: 6-8-10, the doubled 3-4-5 triangle, with area and perimeter 24. If you set 1 over 2ab = a + b + c and use a² + b² = c² to eliminate c, the condition simplifies to (a − 4)(b − 4) = 8, and the only factor pairs of 8 give legs 5 and 12, or 6 and 8.

Second, the altitude to the hypotenuse is not a whole number, even though all three sides are. That is normal: the altitude of a right triangle is ab/c, and 13 does not divide 5 × 12 = 60. The two pieces it cuts from the hypotenuse follow the geometric mean theorem: 25 over 13 × 144 over 13 = (60 over 13)².

Why 5-12-13 is a primitive triple

A Pythagorean triple is primitive when its three numbers have no common factor bigger than 1. Primitive triples are the "originals"; every other triple is a scaled copy of one of them. 5-12-13 is primitive because any common factor would have to divide 5, and 5 is prime, but 12 is not a multiple of 5. So the only common factor is 1. In particular, 5-12-13 is not a multiple of 3-4-5; it is a genuinely different shape, as its angles show.

Euclid's formula produces it from m = 3 and n = 2:

Because 3 and 2 have no common factor and one of them is even, the formula is guaranteed to give a primitive triple. The 5-12-13 triangle also shows the patterns that every primitive triple follows: exactly one leg is even (12), and it is a multiple of 4; the hypotenuse is odd (13); one leg is a multiple of 3 (12); and one side is a multiple of 5 (5). In the tree of primitive Pythagorean triples, 5-12-13 is one of the three "children" of 3-4-5, alongside 20-21-29 and 8-15-17.

Multiples of the 5-12-13 triangle

Multiply every side by the same number k and you get a similar triangle with the same angles: (5k)² + (12k)² = 25k² + 144k² = 169k² = (13k)². The perimeter becomes 30k and the area 30k².

Spotting a disguised multiple saves a lot of arithmetic. If a problem gives legs 15 and 36, divide by 3, recognize 5 and 12, and the hypotenuse is 3 × 13 = 39 without squaring anything. One curiosity: 65 appears as the hypotenuse of four different triples: 25-60-65 (5-12-13 times 5), 39-52-65 (3-4-5 times 13), and the primitive triples 16-63-65 and 33-56-65. That happens because 65 = 5 × 13 and both 5 and 13 are hypotenuses themselves.

Worked examples

For more practice, the word problems page uses these triples in real settings.

The 3-4-5, 5-12-13, 7-24-25 family

In both 3-4-5 and 5-12-13, the hypotenuse is exactly one more than the long leg. If c = b + 1, the theorem becomes

So the square of the short leg must split into two consecutive whole numbers: 25 = 12 + 13. That forces a to be odd (the sum of two consecutive numbers is always odd), so write a = 2n + 1. Then b + c = a² = 4n² + 4n + 1, and with c = b + 1:

Every whole number n ≥ 1 gives a triple, and the argument shows there are no others with c = b + 1. Each one is primitive, because b and c are consecutive and consecutive numbers never share a factor.

Angles are rounded to the nearest hundredth. Two patterns jump out. Every odd number from 3 up is the short leg of exactly one triangle in the family, so there are infinitely many primitive triples. And the inradius (a + b − c)/2 simplifies to (2n + 1 − 1)/2 = n: the 3-4-5 triangle has inradius 1, 5-12-13 has 2, 7-24-25 has 3, and so on.

This "start from an odd number" rule is old. The Greek philosopher Proclus, writing in the 5th century CE, credits it to Pythagoras, and credits Plato with a companion rule that starts from an even number and gives triples with c = b + 2, such as 8-15-17 and 12-35-37. Attributions made a thousand years after the fact are hard to check, but the rules themselves are correct, and both are special cases of Euclid's formula (m = n + 1 gives the odd family). To explore beyond them, try the Pythagorean triples generator or browse the list of Pythagorean triples.