The 3-4-5 Triangle

A 3-4-5 triangle is a right triangle whose sides are 3, 4 and 5 units long, or any multiple of those, such as 6-8-10. It is a right triangle because 3² + 4² = 9 + 16 = 25 = 5². Its angles are 90°, about 36.87° and about 53.13°, and it is the smallest Pythagorean triple.

The two shorter sides, 3 and 4, are the legs: they meet at the right angle. The longest side, 5, is the hypotenuse. Small whole numbers make it the triangle builders use to square corners and the first example most students meet of the Pythagorean theorem.

Why 3, 4 and 5 make a right triangle

Build a square on each side of the triangle and cut it into unit squares. The square on the side of length 3 holds 9 unit squares, the square on the side of length 4 holds 16, and the square on the hypotenuse holds 25. The two smaller squares together contain exactly as many unit squares as the big one.

That is the Pythagorean theorem in its original, geometric form. It also works in reverse: the converse of the Pythagorean theorem says that if three lengths satisfy a² + b² = c², the triangle they form has a right angle. So any triangle with sides 3, 4 and 5 is guaranteed to be a right triangle, with the 90° angle between the sides of length 3 and 4. You don't need a protractor: the side lengths settle it.

3-4-5 triangle angles

One angle is exactly 90°. The other two are not round numbers, but trigonometry gives them quickly. Call the angle opposite the side of length 3 angle A, and the angle opposite the side of length 4 angle B. The tangent of an angle is the opposite side divided by the adjacent side:

Both angles are rounded to the nearest hundredth of a degree, and they add up correctly: 36.87° + 53.13° + 90° = 180°. The sines and cosines, unlike the angles, are exact and tidy, which is why the 3-4-5 triangle appears in so many trigonometry problems:

Properties of the 3-4-5 triangle

Every measurement of the triangle follows from its three sides. These are the ones that come up most often.

The inradius is the radius of the largest circle that fits inside the triangle. For every right triangle it equals r = (a + b − c)/2, which here is exactly 1. The 3-4-5 triangle is in fact the only whole-number right triangle with inradius 1. Setting a + b − c = 2, substituting c = a + b − 2 into a² + b² = c² and simplifying gives (a − 2)(b − 2) = 2, and the only way to write 2 as a product of positive whole numbers is 1 × 2. So the legs must be 3 and 4.

The circumradius, the radius of the circle through all three corners, is 2.5 because the hypotenuse of a right triangle is always a diameter of that circle (Thales' theorem). The altitude from the right angle to the hypotenuse is 2.4, and it splits the triangle into two smaller copies of itself: 1.8-2.4-3 and 2.4-3.2-4, which are the 3-4-5 triangle scaled by 0.6 and by 0.8. That splitting is the idea behind the similar triangles proof and the geometric mean theorem.

Multiples of the 3-4-5 triangle

Multiply all three sides by the same positive number k and you get another right triangle, because

The new triangle is similar to the original, a scaled copy, so it has exactly the same angles: 90°, about 36.87° and about 53.13°. The number k doesn't have to be whole; 1.5-2-2.5 and 0.3-0.4-0.5 are 3-4-5 triangles too.

The perimeter grows in step with k, but the area grows with k²: doubling the sides multiplies the area by 4, tripling multiplies it by 9. Look at the 6-8-10 row: its area and perimeter are both 24. Only one other whole-number right triangle shares that property, the 5-12-13 triangle, whose area and perimeter are both 30. (Setting ab/2 = a + b + c leads to (a − 4)(b − 4) = 8, which allows only these two.)

The 3-4-5 triple is primitive: its three numbers have no common factor. All of its multiples are not. In Euclid's formula it is the very first case, m = 2 and n = 1: m² − n² = 3, 2mn = 4 and m² + n² = 5. For many more triples, see the list of Pythagorean triples.

The 3-4-5 rule for square corners

The 3-4-5 rule, sometimes called the 3-4-5 method, is how carpenters, bricklayers and landscapers check that a corner is exactly 90° without a giant set square. It is the converse of the Pythagorean theorem put to work: if the three distances measure 3, 4 and 5, the corner must be square.

1. Mark 3 units along one side of the corner, measured from the corner point. Any unit works: feet, yards or meters.
2. Mark 4 units along the other side, in the same unit.
3. Measure the diagonal between the two marks. If it is exactly 5 units, the corner is 90°.
4. Adjust and re-measure. A diagonal longer than 5 means the corner is wider than 90° (obtuse); a shorter one means it is narrower (acute). This is the same test described on the acute, right or obtuse page.

Use the biggest multiple that fits, such as 6-8-10 or 9-12-15 feet for a foundation, because a larger triangle makes the same measuring error matter less. With legs of 3 and 4 feet, a diagonal that is 1/8 inch too long means a corner of about 90.25°, which puts a 20-foot wall about an inch out of line at its far end. With legs of 9 and 12 feet, the same 1/8-inch slip gives about 90.08°. (Angles from the law of cosines, rounded to the nearest hundredth.)

The same triangle hides in roof framing. A roof that rises 9 inches for every 12 inches of level run (a "9-in-12" pitch) forms 9-12-15 triangles, so each foot of run needs 15 inches of rafter and the roof slopes at about 36.87°. The rafter length calculator handles every other pitch, and the Pythagorean theorem in construction covers foundations, decks and stairs in detail.

Why 3-4-5 is the only triple of consecutive numbers

3, 4 and 5 are consecutive whole numbers. Could another right triangle have consecutive sides, say 20-21-22? No, and a few lines of algebra prove it for every possible size at once. Call the middle side n, so the sides are n − 1, n and n + 1. The longest side, n + 1, must be the hypotenuse:

So n = 0 or n = 4. A side can't have length 0, so n = 4 and the sides are 3, 4 and 5. This is a proof, not a search result: there is no larger example waiting to be found.

The same argument works for any equally spaced sides. Write them as n − d, n and n + d, with a positive gap d:

The sides are then 3d, 4d and 5d. In words: every right triangle whose sides are equally spaced is a scaled 3-4-5 triangle, even when the gap d is a fraction or an irrational number.

A short history of the 3-4-5 triangle

The 3-4-5 triangle was in use long before Pythagoras (about 570–495 BCE), in several cultures. Here is what the evidence supports, and where it runs out.

• Babylon, about 1800 BCE. The clay tablet Plimpton 322 lists fifteen rows of numbers belonging to right triangles with whole-number sides. Row 11 gives a short side of 45 and a diagonal of 75, which belong to the 45-60-75 triangle: 3-4-5 scaled by 15.
• India, roughly 800–200 BCE. The Sulba Sutras, manuals for building Vedic fire altars with cords and stakes, use 3-4-5 and other triples such as 5-12-13 to lay out right angles. Their dates are uncertain.
• China, Han dynasty. The Zhoubi Suanjing, compiled roughly 2,000 years ago, opens with a dialogue that explains the right-triangle rule using a base of 3, a height of 4 and a diagonal of 5. See the Gougu theorem.
• Rome, 1st century BCE. The architect Vitruvius, in Book IX of De architectura, credits Pythagoras with making a right angle from rods 3, 4 and 5 feet long, and recommends the same proportions for laying out staircases.
• Egypt: a popular but unproven story. You will often read that Egyptian "rope-stretchers" (land surveyors) used a rope with 12 equally spaced knots to make 3-4-5 right angles for temples and pyramids. No ancient text or picture shows this. The idea began as a suggestion by the German historian Moritz Cantor in the late 19th century, and historians still disagree about it. The Egyptian rope-stretchers page weighs the evidence.

Using one triangle is different from proving the theorem for every right triangle. For who proved what, and when, see the history of the Pythagorean theorem.