Pythagorean Theorem Calculator
Solve a right triangle with a² + b² = c²: find the hypotenuse from two legs, or a missing leg from the hypotenuse, with the angles, area, and perimeter.
Comment utiliser cette calculatrice
- 1Choose whether you're finding the hypotenuse or a missing leg.
- 2Enter the two sides you know.
- 3Read the missing side, computed with a² + b² = c².
- 4See the triangle's angles, area, and perimeter alongside.
Comment ça marche
Pythagorean theorem
a² + b² = c² (right triangles only) hypotenuse c = √(a² + b²) missing leg b = √(c² − a²) c is always the longest side, opposite the right angle
The Pythagorean theorem relates the three sides of a right triangle: the square of the hypotenuse — the side opposite the right angle, and always the longest — equals the sum of the squares of the other two sides, the legs. Written a² + b² = c², it lets you find any one side from the other two. To find the hypotenuse, add the squares of the legs and take the square root; to find a leg, subtract the square of the known leg from the square of the hypotenuse and take the square root. The theorem applies only to right triangles, and finding a leg only works when the hypotenuse genuinely is the longest side, since a leg can never exceed it. From the three sides, the two non-right angles follow from basic trigonometry, and the area is simply half the product of the legs.
Exemple détaillé
A right triangle with legs of 3 and 4 has a hypotenuse of √(3² + 4²) = √(9 + 16) = √25 = 5 — the famous 3-4-5 triangle. Its angles are about 36.87° and 53.13°, its area is (3 × 4) ÷ 2 = 6, and its perimeter is 3 + 4 + 5 = 12.
Pythagorean Theorem Calculator : le guide complet
The most famous theorem in mathematics
The Pythagorean theorem is probably the single best-known result in all of mathematics, and for good reason: it is simple to state, easy to use, and astonishingly useful. It says that in any right triangle, the square built on the hypotenuse has exactly the same area as the two squares built on the legs combined. Named for the ancient Greek mathematician Pythagoras, the relationship was actually known and used far earlier by Babylonian and Egyptian builders, who exploited it to lay out right angles long before it was formally proved.
Its enduring importance comes from connecting length and area in a way that lets you compute distances you cannot measure directly. Whenever a problem contains a right angle — and right angles are everywhere, from buildings to screens to coordinate grids — the theorem provides a bridge between the sides. That single idea, that the sides of a right triangle are locked together by a² + b² = c², underlies an enormous amount of practical geometry, trigonometry, and physics.
Pythagorean triples and the 3-4-5 triangle
Certain right triangles have sides that are all whole numbers, and these are called Pythagorean triples. The most famous is 3-4-5, because 3² + 4² = 9 + 16 = 25 = 5². Others include 5-12-13, 8-15-17, and 7-24-25, along with every multiple of these, like 6-8-10. These special triangles have been prized since antiquity precisely because they give exact right angles without any measurement of angles at all — a fact builders still use.
The 3-4-5 triangle in particular is a carpenter's and mason's staple for squaring corners. By measuring 3 units along one edge and 4 along another, then adjusting until the diagonal between the marks is exactly 5, you guarantee a perfect 90° corner using only a tape measure. Larger multiples like 6-8-10 or 9-12-15 make the method more accurate over the distances of a real building. This is the Pythagorean theorem doing everyday work, turning a piece of pure number theory into a reliable practical tool.
From triangles to distance and beyond
The Pythagorean theorem is the foundation of the distance formula in coordinate geometry. The straight-line distance between two points is just the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps between them — so the distance is the square root of the sum of the squared differences in the coordinates. Every time a map app computes a straight-line distance, a game engine measures how far apart two objects are, or a physicist combines perpendicular components of a force or velocity, the Pythagorean theorem is quietly at work.
It also generalises beyond flat triangles. In three dimensions, the diagonal of a box is found by applying the theorem twice, and the idea extends to any number of dimensions in the mathematics of vectors. It underpins trigonometry, where the relationship sin² + cos² = 1 is the Pythagorean theorem in disguise, and it appears throughout physics wherever quantities combine at right angles. What looks like a humble fact about triangles is in truth one of the deepest and most far-reaching ideas in mathematics, and mastering it in its simplest form opens the door to all of that.
Questions fréquentes
What is the Pythagorean theorem?
For any right triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the legs. It lets you find any side from the other two: the hypotenuse is √(a² + b²), and a missing leg is √(c² − a²).
How do I find the hypotenuse?
Square both legs, add them, and take the square root: c = √(a² + b²). For legs of 3 and 4, that's √(9 + 16) = √25 = 5. The hypotenuse is always the longest side and lies opposite the right angle.
How do I find a missing leg?
Rearrange to b = √(c² − a²): square the hypotenuse, subtract the square of the known leg, and take the square root. This only has a solution when the hypotenuse is longer than the known leg, since the hypotenuse is always the longest side of a right triangle.
What is a Pythagorean triple?
A right triangle whose sides are all whole numbers, like 3-4-5, 5-12-13, or 8-15-17, along with their multiples. They give exact right angles without measuring angles, which is why builders use the 3-4-5 method to square corners with only a tape measure.