The Pythagorean theorem is the oldest theorem in mathematics that is still exactly true: a² + b² = c², for every right triangle, everywhere, including in countries that have not been invented yet. It has been proven in more ways than any other result in the subject, and it is the first step from arithmetic to geometry.
The statement
In a right triangle — one with a 90° angle — the sides touching that angle are the legs, a and b, and the side opposite it is the hypotenuse, c, always the longest:
a² + b² = c²
The Pythagorean theorem calculator solves for the hypotenuse or a missing leg, and will tell you whether a given triangle would be right-angled at all.
Three proofs, because the proof is the point
The area argument. Draw a square of side (a + b) on each side of the right triangle. The big square's area is (a + b)² = a² + 2ab + b². The two small squares have area a² and b², and the triangle appears twice inside the big square, contributing 2ab — which cancels, leaving a² + b² = c².
The rearrangement argument. Build a (c × c) square from four copies of the triangle plus one small square of side (b − a). Its area is c², and also 4 × (ab/2) + (b − a)² = 2ab + b² − 2ab + a². The 2ab terms cancel again, and the result is c² = a² + b².
The similarity argument. Draw an altitude from the right angle to the hypotenuse. The two smaller triangles are similar to the original and to each other, so the proportions of their sides give (a/c) = (a'/a), which rearranges to a² = c·a'. Doing the same for b gives b² = c·b', and since a' + b' = c, adding yields a² + b² = c².
Three quite different arguments, one result. That a theorem with this many clean proofs is a sign of how constrained it is.
Worked example: finding a missing leg
A wall is 3 ft tall and a diagonal brace must reach 5 ft horizontally. Here c = 5 and b = 3, so:
a = √(c² − b²) = √(25 − 9) = √16 = 4 ft
The neat part is the direction of the subtraction. The hypotenuse is always the thing you subtract from, because it is always the largest side — putting it on the other side of the equation gives a negative number under the root, which is the calculator's way of saying "no such triangle".
Worked example: legs of 6 and 8
c = √(6² + 8²) = √(36 + 64) = √100 = 10
The 6-8-10 triangle, which scales the 3-4-5 by a factor of two. That family — 3-4-5, 6-8-10, 9-12-15, and the half-integer variants — is the practical core of the theorem, and it is the basis of the 0.6 × 0.8 × 1.0 right triangle used for roof pitch, stair rise-and-run, and A-frame geometry. A 45° diagonal is the simplest case: legs equal, so the hypotenuse is the leg times √2 ≈ 1.414.
Using it as a right-angle test
The theorem is bidirectional. If you have three lengths and a² + b² = c² exactly, the triangle is right-angled. Sort the sides first — the longest must be c.
Sides 7, 24, 25: 49 + 576 = 625 = 25². Right-angled, and it is a 7-24-25 triangle.
Sides 5, 6, 7: 25 + 36 = 61 ≠ 49. Not right-angled; the longest angle is acute-ish and the triangle is obtuse, because 61 > 49 tells you the side opposite the last angle is longer than a right triangle's would be.
On a construction site this is a real technique. Tape out where the post should go, measure three sides, and the arithmetic confirms the corner is square — faster and more certain than eyeballing a diagonal.
The box diagonal, in two steps
To find the space diagonal of a rectangular box, apply the theorem twice. Treat the footprint diagonal as one leg and the height as the other:
Footprint of a 40 × 30 cm box: √(40² + 30²) = √(1600 + 900) = √2500 = 50 cm.
With a height of 30 cm: total diagonal = √(50² + 30²) = √(2500 + 900) = √3400 ≈ 58.3 cm.
Which is just the three-term version d = √(x² + y² + z²) reached in two steps — the same form as the distance formula in three dimensions.
What it does not apply to
The theorem needs a right triangle. For an oblique triangle the closest tool is the law of cosines, c² = a² + b² − 2ab·cos C, which reduces to Pythagoras exactly when C = 90° and cos C = 0. That reduction is the quickest way to remember the law of cosines: Pythagoras is the special case.
It also does not apply on a sphere. Curved geometry changes the rules for both angles and distances, and a triangle on the Earth's surface has interior angles summing to more than 180° — which is why great-circle routes bend polewards on a flat map.
The Pythagorean family: other integer triples
The 3-4-5 is only the smallest. The full family of integer right triangles comes from parameters m and n with m > n:
a = m² − n², b = 2mn, c = m² + n²
With m = 2, n = 1: (3, 4, 5). With m = 3, n = 2: (5, 12, 13). With m = 4, n = 1: (15, 8, 17). With m = 4, n = 3: (7, 24, 25). With m = 5, n = 2: (21, 20, 29).
The reason this family exists is more interesting than the list. Integer right triangles correspond exactly to rational points on the unit circle, and the parameterisation above is a consequence of the number theory of the sum of two squares. The practical upshot is simple: if a triangle has integer sides and a right angle, its sides are almost certainly drawn from this family — a useful check when a dimension looks odd.
The primitive triples (those with no common factor) are exactly the ones where m and n are coprime and of opposite parity; every other integer triple is a multiple of one of these.
Scale factor, similarity and the real uses
Because the theorem is about a ratio, it transfers across scale. A 30-40-50 triangle is the 3-4-5 doubled, and its angle is identical. This is why surveyors work with scaled drawings, why a scaled architectural plan still carries every geometric relationship of the building, and why a triangle measured with a tape at 1:100 scale behaves exactly like the real thing.
Three practical applications worth knowing by name:
- Diagonal bracing: a 3 m tall wall needing a 5 m diagonal is braced horizontally at 4 m — the same calculation in reverse.
- Screen and box sizing: the 16:9 display diagonal is the hypotenuse, giving width = d × 16/√337 and height = d × 9/√337.
- Right-angle checking on site: tape out 3 m along one edge, 4 m up the next, and measure the 5 m diagonal. If it is 5 m, the corner is square.
Frequently asked questions
How do you find the hypotenuse of a right triangle?
Square both legs, add them, then take the square root: c = √(a² + b²). For legs of 3 and 4 this gives c = √25 = 5.
How do you find a missing leg?
Square the hypotenuse, subtract the square of the known leg, and take the root: a = √(c² − b²). The hypotenuse must always be the value you subtract from.
How do I know if a triangle is right-angled?
Sort the three sides shortest to longest and check whether a² + b² equals c². If it matches exactly, the angle opposite the longest side is 90°.
Does the theorem work in three dimensions or on a sphere?
In 3D, extend it by adding a third squared term. On a sphere, no — the geometry is curved, and angles and distances follow different rules entirely.