Find the hypotenuse: legs 3 and 4
- Input
- a = 3, b = 4, Find = hypotenuse c
- Working
- c = √(a² + b²) = √(9 + 16) = √25 = 5. The result is a whole number because 3-4-5 is a Pythagorean triple — the first row of the table above.
- Result
- Hypotenuse c = 5
Pythagorean Theorem Calculator
Find any side of a right-angled triangle from the other two — hypotenuse or leg — with the squaring and root steps shown and classic triples recognised. Runs entirely in your browser — your numbers never leave this device.
Solve for
Hypotenuse c
5
c = √(3² + 4²) = √25 = 5
- Side a
- 3
- Side b
- 4
- Hypotenuse c
- 5
- Perimeter
- 12
- Area
- 6
How it works
For a right-angled triangle with legs a and b and hypotenuse c: c = √(a² + b²) | a = √(c² − b²) | b = √(c² − a²)
| Leg a | Leg b | Hypotenuse c | Check: a² + b² = c² |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
Pythagoras' theorem connects the three sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides, a² + b² = c². The hypotenuse is always the side opposite the right angle and always the longest side — identifying it correctly is the whole setup, and it decides which version of the formula you need. Enter the two sides you know, tell the calculator which side is missing, and it shows the squaring, the addition or subtraction, and the final square root, so each step can be followed on paper.
The direction of the working flips depending on what is missing. Finding the hypotenuse adds the squares: legs 3 and 4 give √(9 + 16) = √25 = 5. Finding a leg subtracts instead: the known leg's square comes off the hypotenuse's square before rooting, because the leg is only part of what the hypotenuse's square contains. A built-in sanity check guards the commonest slip: a leg must come out shorter than the hypotenuse, so if a subtraction ever heads towards a negative number — a 'leg' longer than the hypotenuse — the calculator flags an impossible triangle rather than rooting a negative.
Some right triangles have all three sides whole, and they are worth recognising on sight: the Pythagorean triples in the table below, starting with 3-4-5. Triples scale — multiply every side by the same number and the triangle still works, so 6-8-10 and 9-12-15 are just 3-4-5 in disguise — which makes them a fast estimation check for non-triple problems. The theorem itself applies only when there is a genuine right angle; for other triangles the law of cosines is the tool, and no amount of a² + b² will substitute for it.
Worked examples
Find a leg: hypotenuse 13, one leg 5
- Input
- c = 13, a = 5, Find = leg b
- Working
- b = √(c² − a²) = √(169 − 25) = √144 = 12. Note the subtraction: the leg comes from what remains of the hypotenuse's square after the other leg's square is removed. This is the 5-12-13 triple.
- Result
- Other leg b = 12
A ladder against a wall
- Input
- Ladder (hypotenuse) = 5 m, Foot of ladder from wall (leg) = 3 m, Find = height up the wall
- Working
- The ladder, the wall and the ground form a right-angled triangle with the ladder as hypotenuse. Height = √(5² − 3²) = √(25 − 9) = √16 = 4. So a 5 m ladder planted 3 m out reaches exactly 4 m up the wall — the 3-4-5 triple doing everyday work.
- Result
- The ladder reaches 4 m up the wall
Frequently asked questions
Which side is the hypotenuse?
The side opposite the right angle, and always the longest of the three sides. It is labelled c in the standard formula a² + b² = c². If you know the longest side of your triangle, that is the hypotenuse; if a calculated 'leg' ever comes out longer than it, the sides were entered in the wrong roles — a leg longer than the hypotenuse is geometrically impossible.
Why do I subtract to find a leg but add to find the hypotenuse?
Because of what the equation says: a² + b² = c² means each leg's square is only part of the hypotenuse's square. Solving for the hypotenuse keeps the sum together, c = √(a² + b²). Solving for a leg means isolating its square first — b² = c² − a² — and only then taking the root. Adding when you should subtract is the most common Pythagoras error, and it always produces a 'leg' larger than the hypotenuse, which is your warning sign.
What is a Pythagorean triple?
A set of three whole numbers that satisfy the theorem exactly, like 3-4-5, 5-12-13 and 20-21-29 from the table above. Some are primitive (the three numbers share no common factor, as with 20-21-29) and some are scalings of smaller triples — 6-8-10 is 3-4-5 doubled. Recognising a triple lets you answer without any rooting at all.
Does the theorem work for triangles without a right angle?
No. The equality a² + b² = c² holds only when the angle between a and b is exactly 90°. In fact the comparison works as a test: if a² + b² is greater than c² the angle opposite c is acute, and if it is less, that angle is obtuse. For non-right triangles generally, the law of cosines extends the theorem by adding a correction term involving the angle.
What can I use the Pythagorean theorem for in real life?
Anywhere a right angle and a diagonal meet. Checking a frame is square (its diagonal should match the value from its sides), the distance across a rectangular pitch or screen — a 6 by 8 rectangle has a diagonal of √(36 + 64) = √100 = 10 — ladder reach, guy-rope lengths, and the straight-line distance between two points on a map grid, which is exactly the hypotenuse of the east and north differences.
Are the measurements I enter uploaded anywhere?
No. This calculator runs 100% in your browser — the sides you type are processed locally on your device, nothing is uploaded, saved or tracked, and closing the page clears everything.
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