Find a right triangle's hypotenuse from its two legs, or find a missing leg from the hypotenuse and one leg.
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The Pythagorean theorem is one of the oldest and most famous theorems in geometry, stating that the square of the hypotenuse's length in any right triangle equals the sum of the squares of the other two sides' lengths (a² + b² = c²). This calculator allows using the theorem in two directions: the first is calculating the hypotenuse's length directly if the two leg lengths are known, and the second is calculating an unknown leg's length if the hypotenuse and the other leg's length are known. This theorem is used in practice in construction, navigation, and calculating the shortest diagonal distance between two points on a rectangular grid. The theorem applies exclusively to right triangles — triangles containing exactly one 90-degree angle — which is why the more general Law of Cosines is needed whenever a triangle's angles are not known to include a right angle. The same equation also underlies the well-known 3-4-5 rule used on construction sites to check that a corner is genuinely square, since 3² + 4² equals 5² exactly.
Few results in all of mathematics are as immediately recognizable or as broadly useful as the Pythagorean theorem: in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Despite being credited to the ancient Greek mathematician Pythagoras, evidence suggests the relationship was known and used practically by Babylonian and Egyptian builders centuries earlier, underscoring just how fundamentally useful — and how early it was discovered — this single equation turned out to be.
The theorem works in two directions equally well. Given the two shorter sides (called the 'legs') of a right triangle, squaring and adding them, then taking the square root, gives the hypotenuse directly. Given the hypotenuse and one leg, the theorem can be rearranged — subtracting the known leg's square from the hypotenuse's square, then taking the square root — to find the missing leg instead.
Construction relies on this theorem constantly through a practical technique called the '3-4-5 rule': because 3² + 4² = 5² (9 + 16 = 25), a triangle with sides in the ratio 3:4:5 is guaranteed to contain a perfect right angle. Builders use this ratio (or any multiple of it, such as 6-8-10) to square up foundations, walls, and door frames accurately using nothing more than a tape measure, without needing a specialized right-angle tool.
Navigation and surveying use the theorem to calculate direct, straight-line distances when only north-south and east-west displacement are known — exactly the situation when a ship or aircraft has traveled a certain distance north and a certain distance east, and the shortest direct return distance needs to be calculated. This is precisely the same logic used in the distance formula between two coordinate points, which is itself the Pythagorean theorem applied to horizontal and vertical coordinate differences.
Beyond construction and navigation, the theorem underlies calculating a television or monitor's diagonal screen size from its width and height, determining the correct length of a ladder needed to safely reach a given height at a safe base distance from a wall, and calculating the diagonal bracing needed to reinforce a rectangular frame — any situation, in short, where a diagonal length needs to be derived from two measurements taken at a right angle to each other.
No, it only applies to right triangles — triangles that contain exactly one 90° angle.
The hypotenuse is always the longest side of a right triangle, located directly opposite the right angle.
That combination is impossible in a real right triangle, since the hypotenuse must always be the longest side, so the calculator will ask you to recheck your values.