5-12-13 Triangle: Formula, Angles & Worked Examples

A 5 12 13 triangle is a right triangle, and one line of arithmetic proves it: 5² + 12² = 25 + 144 = 169, and 13² = 169, so 5² + 12² = 13². Because the three sides satisfy the Pythagorean relationship exactly, the angle opposite the side of length 13 is exactly 90°. The 13 is therefore always the hypotenuse, never a leg, and the two legs of 5 and 12 meet at the right angle.

Figure 1

Why 5-12-13 is a primitive Pythagorean triple

A Pythagorean triple is any set of three positive whole numbers a, b, c with a² + b² = c². A triple is primitive when those three numbers share no common factor greater than 1, which means it is not a scaled copy of a smaller triple.

Now test primitivity by taking the greatest common divisor. The divisors of 5 are 1 and 5. The divisors of 12 are 1, 2, 3, 4, 6 and 12. The only value on both lists is 1, so gcd(5, 12) = 1, and gcd(1, 13) = 1. The greatest common divisor of all three numbers is 1, so 5-12-13 is primitive.

Primitivity means 5-12-13 is the smallest member of its own infinite family of similar triangles, and that it is a genuinely different shape from 3-4-5 rather than a rescaling of it: the 3-4-5 triangle rule gives acute angles near 36.87° and 53.13°, while 5-12-13 is far more slender. It is the second smallest primitive triple by hypotenuse, behind 3-4-5 and ahead of 8-15-17. The wider family, including how primitives are generated, is covered in the guide to Pythagorean triples.

The exact angles: 22.62° and 67.38°

The two acute angles follow from the tangent ratio, because in a right triangle the tangent of an acute angle is the opposite leg divided by the adjacent leg.

Computed to seven decimal places:

• Angle opposite the side of length 5: arctan(5/12) = 22.6198649°, normally rounded to 22.62° (22° 37′ to the nearest arcminute)
• Angle opposite the side of length 12: arctan(12/5) = 67.3801351°, normally rounded to 67.38° (67° 23′)
• Angle opposite the side of length 13: exactly 90°

The two acute angles must sum to 90°, and they do: 22.6198649 + 67.3801351 = 90.0000000. Using the rounded pair gives 22.62 + 67.38 = 90.00 as well, which is a quick sanity check you can run in your head.

Area, perimeter and the altitude to the hypotenuse

The two legs meet at the right angle, so they are the base and the height and the area needs no extra construction.

The area and the perimeter both come out as 30, a coincidence of the numbers rather than a geometric law: 30 square units and 30 linear units are not comparable quantities.

The altitude to the hypotenuse

Drop a perpendicular from the right angle onto the hypotenuse. Its length h can be found by computing the area twice, once from the legs and once from the hypotenuse and h:

Check it by recomputing the area from the hypotenuse: ½ à - 13 à - 60/13 = ½ à - 60 = 30 square units, which matches the leg calculation exactly. Keep the answer as the fraction 60/13 wherever you can, since 4.62 is only a two-decimal approximation.

The foot of that altitude splits the hypotenuse into two segments, p adjacent to leg a and q adjacent to leg b, given by p = a²/c and q = b²/c:

A second independent check uses the geometric mean relationship h² = pq: (60/13)² = 3600/169 ≈ 21.3018, and (25/13) à - (144/13) = 3600/169 ≈ 21.3018. The two agree, which is the right triangle altitude theorem applied to this triple.

Figure 2

Scaled versions of the 5-12-13 triangle

Multiplying all three sides by the same positive number k gives another right triangle, because (5k)² + (12k)² = 25k² + 144k² = 169k² = (13k)². Every multiple of 5-12-13 is therefore a valid Pythagorean triple, and none of them is primitive.

Two patterns in that table are worth holding on to. The perimeter scales by k, so it goes 30, 60, 90, 120, 150. The area scales by k², so it goes 30, 120, 270, 480, 750, which is 30 à - 1, 30 à - 4, 30 à - 9, 30 à - 16, 30 à - 25.

The angles do not change at all. Every triangle in the table has acute angles of arctan(5/12) = 22.62° and arctan(12/5) = 67.38°, because the ratio of the legs is 5:12 in every row and the tangent depends only on that ratio. Scaling produces similar triangles, identical in shape and different only in size. For k = 3 the check is 15/36 = 5/12, so arctan(15/36) is the same number as arctan(5/12).

Scaling also works with k less than 1 and with non-integer k. Halving gives 2.5, 6, 6.5, which satisfies 6.25 + 36 = 42.25 = 6.5².

How to recognise a 5-12-13 in an exam question

A question almost never says “this is a 5-12-13”. It gives you two numbers and expects you to spot the pattern instead of reaching for a calculator. Three disguises are common.

Disguise 1: the sides are scaled. Divide the numbers you were given by their common factor and see whether 5, 12 and 13 fall out. Sides of 45 and 108 share a factor of 9, and 45 ÷ 9 = 5 with 108 ÷ 9 = 12, so the hypotenuse is 13 à - 9 = 117. A faster test when a 13-family number is present: divide the hypotenuse by 13. If 65 is the hypotenuse, 65 ÷ 13 = 5, so the legs are 5 à - 5 = 25 and 12 à - 5 = 60.

Disguise 2: the sides are listed in a different order. A triangle described as having sides 39, 15 and 36 is a 5-12-13 with k = 3, but written out of order. Sort the three values first: 15, 36, 39. The largest is the hypotenuse, always. Then check 15² + 36² = 225 + 1296 = 1521, and 39² = 1521.

Disguise 3: you are given a leg and the hypotenuse. If the hypotenuse is 52 and one leg is 20, divide both by 4 to get 13 and 5, so the missing leg is 12 à - 4 = 48. Confirm with the theorem: 20² + 48² = 400 + 2304 = 2704 = 52².

Worth memorising: hypotenuses of 13, 26, 39, 52 and 65 are all 5-12-13 candidates. When the numbers refuse to cooperate, fall back on the theorem or let the right triangle calculator do the arithmetic.

Figure 3

Exact trig ratios for the 5-12-13 triangle

Because the sides are whole numbers, every trig ratio in this triangle is an exact fraction. Call the smaller acute angle A (opposite the leg of 5) and the larger acute angle B (opposite the leg of 12).

Every decimal in that column was checked against the angle itself: sin(22.6198649°) = 0.384615, cos(22.6198649°) = 0.923077 and tan(22.6198649°) = 0.416667, matching 5/13, 12/13 and 5/12 to six decimal places.

Notice the crossover: sin A equals cos B and cos A equals sin B, because A and B are complementary. Notice too that tan A à - tan B = (5/12) à - (12/5) = 1. The identity sin²A + cos²A = 25/169 + 144/169 = 169/169 = 1 is the same 5² + 12² = 13² arithmetic wearing a different hat. If the naming of opposite, adjacent and hypotenuse is what trips you up, the SOHCAHTOA method sets out the labelling rules.

Six worked examples

The square root comes out whole, which is the signature of a Pythagorean triple. Check by substituting back: 5² + 12² = 169 and 13² = 169.

Spotted faster: 52 ÷ 4 = 13 and 20 ÷ 4 = 5, so this is 5-12-13 with k = 4 and the missing leg is 12 à - 4 = 48. Both routes give 48, and 20² + 48² = 400 + 2304 = 2704 = 52².

Confirm the scale factor on each side independently: 45 ÷ 5 = 9, 108 ÷ 12 = 9 and 117 ÷ 13 = 9. All three agree, so the triangle is similar to 5-12-13 and its acute angles are still 22.62° and 67.38°.

The partner angle is 90° − 67.3801351° = 22.6198649° ≈ 22.62°, and the three angles sum to 67.38 + 22.62 + 90 = 180°. Set the calculator to degree mode before using arctan, or you will read 1.176 radians instead.

Cross-check against the scaling rule. This is 5-12-13 with k = 3, so the area should be 30 à - 3² = 30 à - 9 = 270 m² and the perimeter should be 30 à - 3 = 90 m. Both match.

Verify with the area: ½ à - 26 à - 120/13 = ½ à - 240 = 120 square units, and ½ à - 10 à - 24 = 120 square units. The hypotenuse segments are p = 10²/26 = 100/26 = 50/13 ≈ 3.85 and q = 24²/26 = 576/26 = 288/13 ≈ 22.15, and 50/13 + 288/13 = 338/13 = 26, the full hypotenuse.

The numbers 2.5, 6 and 6.5 are 5-12-13 halved (k = 0.5), so the ramp angle is the familiar 22.62° with no new calculation. That slope is far beyond typical accessibility limits of about 4.8°, so the 5-12-13 shape marks a ramp nobody could use. You can check a figure like this before committing it to a drawing.

Common mistakes

Treating 13 as a leg. The most frequent error is writing 5² + 13² = c² and getting c = √194 ≈ 13.93. In a Pythagorean triple the largest number is always the hypotenuse, so 13 belongs alone on the right-hand side. Before substituting anything, sort the three lengths and put the largest into c.

Scaling by adding instead of multiplying. Adding 1 to each side gives 6, 13, 14, which is not a right triangle: 6² + 13² = 36 + 169 = 205, while 14² = 196. The two values differ by 9, so the triangle is not right-angled. Similarity requires every side to be multiplied by the same factor, never shifted by the same amount.

Confusing 5-12-13 with a near miss. Sides of 5, 12 and 14 look close enough to pass a glance, but 5² + 12² = 169 and 14² = 196. The test fails, and the triangle is obtuse: the angle opposite the 14 is arccos((25 + 144 − 196) / (2 à - 5 à - 12)) = arccos(−0.225) ≈ 103.00°, not 90°. A near miss of one unit in the hypotenuse swings the angle by 13 degrees, so never accept a triple because it looks familiar. Run the arithmetic.

Assuming any triangle containing a 13 is a 5-12-13. Legs of 7 and 12 give a hypotenuse of √193 ≈ 13.89, not 13. The triple only applies when both legs match the 5:12 ratio.

Reusing rounded angles. Typing 22.62 into a later sine calculation introduces error that compounds. Keep arctan(5/12) in the calculator’s memory and round once, at the end.

Practice questions

1. A right triangle has legs of 25 cm and 60 cm. Find the hypotenuse, and state the scale factor relative to 5-12-13.
2. A right triangle has a hypotenuse of 39 mm and one leg of 36 mm. Find the third side.
3. A right triangle has legs of 30 and 72. Find its smaller acute angle to two decimal places, without using the inverse tangent of 30/72 directly.
4. A right triangle has sides 20, 48 and 52. Find its area and the altitude drawn to the hypotenuse.

Answer key

1. 65 cm, with k = 5. By the theorem, c = √(25² + 60²) = √(625 + 3600) = √4225 = 65. Confirm the scale factor: 25 ÷ 5 = 5, 60 ÷ 12 = 5 and 65 ÷ 13 = 5.
2. 15 mm. Divide by the common factor 3: 39 ÷ 3 = 13 and 36 ÷ 3 = 12, so this is 5-12-13 with k = 3 and the missing leg is 5 à - 3 = 15. Check directly: √(39² − 36²) = √(1521 − 1296) = √225 = 15.
3. 22.62°. Reduce 30 and 72 by their common factor 6 to get 5 and 12, so the triangle is 5-12-13 with k = 6 and the smaller acute angle is the standard arctan(5/12) = 22.6198649° ≈ 22.62°. (Verification: 30² + 72² = 900 + 5184 = 6084 and 78² = 6084, and arctan(30/72) gives the same 22.6198649°.)
4. Area 480 square units, altitude 240/13 ≈ 18.46. Area = ½ à - 20 à - 48 = 480. Altitude h = ab/c = (20 à - 48)/52 = 960/52 = 240/13 ≈ 18.4615. Check by recomputing the area from the hypotenuse: ½ à - 52 à - 240/13 = ½ à - 960 = 480. ✓

Frequently Asked Questions

Is 5-12-13 a right triangle?

Yes. 5² + 12² = 25 + 144 = 169 and 13² = 169, so the three sides satisfy a² + b² = c² exactly. By the converse of the Pythagorean theorem the angle opposite the side of length 13 is exactly 90°, with no approximation involved.

Is 5-12-13 a primitive Pythagorean triple?

Yes. The greatest common divisor of 5, 12 and 13 is 1, so the triple cannot be reduced to anything smaller and is not a scaled copy of another triple. It is the second smallest primitive triple by hypotenuse, after 3-4-5.

What are the angles of a 5-12-13 triangle?

They are 90°, arctan(5/12) ≈ 22.62° and arctan(12/5) ≈ 67.38°. The 22.62° angle sits opposite the leg of length 5 and the 67.38° angle sits opposite the leg of length 12. The two acute values sum to exactly 90°, as they must in any right triangle.

Is 5-12-13 a special right triangle?

No, not in the usual classroom sense. “Special right triangle” normally means the 30-60-90 and 45-45-90 triangles, whose angles are whole numbers and whose side ratios involve √3 and √2. The 5-12-13 triangle is special in a different way: its sides are whole numbers while its angles are irrational.

Do 10-24-26 and 25-60-65 have the same angles as 5-12-13?

Yes. Every multiple of 5-12-13 has acute angles of 22.62° and 67.38°, because the tangent of each angle depends only on the ratio of the legs, and that ratio stays at 5:12 no matter what you multiply by. Scaling changes side lengths, area and perimeter, but never angles.