Transparent method
The formula
Worked example
Example inputs
How to interpret the result
The theorem only holds when the angle between the two given sides is genuinely ninety degrees. Apply it to a triangle that merely looks roughly square and the hypotenuse comes out wrong, sometimes by a meaningful margin depending on how far the real angle sits from ninety.
Before you rely on it
What to check
Confirm the two known sides actually meet at a right angle, ideally measured, before running the calculation. A framing square or a measured 3-4-5 check is more reliable than eyeballing it.
The common error
Where people go wrong with pythagorean theorem calculator
Applying the theorem to any triangle with a roughly square corner. Off by even a few degrees, the result stops being the hypotenuse of anything real and becomes a number that only looks like one.
Sensitivity evidence
How leg a changes the hypotenuse
| Leg A | Hypotenuse | Triangle area |
|---|---|---|
| 2.4 | 4.7 | 4.8 |
| 2.7 | 4.8 | 5.4 |
| 3worked example | 5 | 6 |
| 3.3 | 5.2 | 6.6 |
| 3.6 | 5.4 | 7.2 |
Every input, tested
Which input moves the hypotenuse most
| Input | Tested from | To | Hypotenuse at each end | Swing |
|---|---|---|---|---|
| Leg B | 3.6 | 4.4 | 4.7 to 5.3 | 0.639 (13%) |
| Leg A | 2.7 | 3.3 | 4.8 to 5.2 | 0.36 (7.2%) |
Two variables at once
Hypotenuse by leg a and leg b
| Leg A \ Leg B | 3.2 | 4 | 4.8 |
|---|---|---|---|
| 2.4 | 4 | 4.7 | 5.4 |
| 2.7 | 4.2 | 4.8 | 5.5 |
| 3 | 4.4 | 5 | 5.7 |
| 3.3 | 4.6 | 5.2 | 5.8 |
| 3.6 | 4.8 | 5.4 | 6 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Leg A | 3 |
| Leg B | 4 |
| Hypotenuse | 5 |
| Triangle area | 6 |
Where the inputs come from
Enter the lengths of the two legs of a right triangle, the sides that meet at the right angle, in the same unit. The hypotenuse is the longest side, opposite the right angle. The theorem says the square of the hypotenuse equals the sum of the squares of the legs. To find a leg instead, use the right triangle calculator. Make sure the triangle really has a right angle before using the theorem.
When the result stops applying
The theorem applies only to right triangles, and for other triangles you need the law of cosines. Measurement errors show up in the result, especially for long distances. Units must match. In three dimensions, the space diagonal of a box extends the formula with a third term.
What to compare it with
Solve for angles and missing legs on the right triangle calculator, general triangles on the law of sines and triangle side pages, and distance on the distance and slope calculator. The rectangular prism page gives space diagonals, and the roof and stair tools in the carpentry category use right triangles.
People also ask
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
What are Pythagorean triples?
Sets of whole numbers that satisfy the theorem, such as 3-4-5, 5-12-13 and 8-15-17, useful for quick checks.
How can I check a corner is square?
Measure 3 units along one side and 4 along the other. If the distance between the marks is 5 units, the corner is a right angle.