The hypotenuse is the longest side of any right triangle and sits directly opposite the 90-degree angle. To find its length, you apply the Pythagorean theorem: c = sqrt(a² + b²), where a and b are the two legs. This calculator takes the two leg lengths as inputs and returns the hypotenuse, both acute angles, the triangle area, and the perimeter. It is the fastest way to work out the diagonal distance between two perpendicular measurements without reaching for a manual square-root calculation.

The Formula: c = sqrt(a² + b²)

The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the two legs. To figure out c, you square both legs, add the results, and take the square root of the sum. Given that both squaring and addition are commutative, it does not matter which leg you call a and which you call b; the hypotenuse is the same regardless of labelling order. The formula works for any consistent unit system: centimetres, metres, feet, or inches all produce valid outputs as long as both inputs share the same unit.

That said, a common source of confusion is applying the formula in reverse. If you have the hypotenuse and one leg and want to find the missing leg, you use a = sqrt(c² - b²) or b = sqrt(c² - a²). This calculator specifically handles the forward direction (both legs known, find c). Our Pythagorean Theorem Calculator handles all three directions and is the better choice when you need to find a missing leg rather than the hypotenuse. The Omni Calculator hypotenuse guide explains the formula and its derivations with diagrams.

Real-World Uses of Hypotenuse Calculations

Hypotenuse calculations appear in any situation where a straight-line diagonal distance is needed from two perpendicular measurements. Screen and monitor sizes are advertised as the diagonal, which is the hypotenuse of the rectangle formed by width and height. A 55-inch screen with a 16:9 ratio has legs of approximately 47.96 inches (width) and 26.98 inches (height), giving a hypotenuse of exactly 55 inches. Cable installers use the same calculation to figure out diagonal conduit runs between two junction boxes at different heights and horizontal distances.

Construction workers carry out the 3-4-5 square check using exactly this formula. Surveyors build up horizontal distances from perpendicular traverse legs when direct measurement is blocked. Roofers calculate rafter length from the horizontal run and vertical rise. As a result, the hypotenuse calculation is one of the most frequently applied mathematical operations in trade work. Our Right Triangle Calculator extends the result by also returning both acute angles, which is useful when pitch or bearing direction matters. The Inch Calculator hypotenuse reference documents six different formula variations for finding the hypotenuse from different input combinations.

Common Hypotenuse Values and Pythagorean Triples

Certain leg combinations produce a whole-number hypotenuse. These are called Pythagorean triples, and the most familiar is 3-4-5: legs of 3 and 4 give a hypotenuse of exactly 5. On top of that, any scaling of a known triple also produces a whole-number result. The table below lists the most common triples encountered in geometry, construction, and standardized tests.

Angles from the Hypotenuse

Once the hypotenuse is known, both acute angles can be calculated. Angle A (opposite leg a) equals arcsin(a/c), and angle B (opposite leg b) equals arcsin(b/c). Equivalently, angle A = arctan(a/b) and angle B = 90 - A. With that in mind, the hypotenuse alone is sometimes the primary output needed, while the angles are secondary; this calculator returns all four derived values so you can narrow down which outputs are relevant to your specific task without switching tools.

The Khan Academy Pythagorean theorem section provides thorough visual explanations of how the hypotenuse relates to the triangle's area, angles, and perimeter, which is useful background for understanding all the outputs this calculator returns.

Verification and Error Checking

To verify a hypotenuse result manually, square the answer and check it equals a² + b². For example, legs of 6 and 8 give c = 10; squaring gives 100, and 36 + 64 = 100 confirms the answer. A quick sense-check is that the hypotenuse must always be greater than either individual leg but less than their sum. If you enter legs of 5 and 5, the hypotenuse should be between 5 and 10; the result of 7.071 passes both bounds. Any computed value outside these bounds indicates a data-entry error worth reviewing before using the result in a real-world calculation.