Use the free Pythagorean Theorem Calculator below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
The Theorem in One Sentence
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is the side opposite the right angle, and it is always the longest side. This one equation, known for more than two thousand years, connects lengths to angles and still underpins construction, navigation, surveying and computer graphics today.
The theorem is a statement about squares, not just about lengths. If you build a square on each side of a right triangle, the area of the square on the hypotenuse equals the combined area of the squares on the other two sides. For the famous 3-4-5 triangle, the squares have areas 9 and 16, and 9 + 16 = 25, which is the square on the hypotenuse of length 5. Seeing the areas makes the equation feel concrete rather than abstract.
How to Use This Calculator
Pick the missing side: a, b or c. Enter the two sides you know, then press Find missing side. If the hypotenuse is missing, the tool computes √(a² + b²). If a leg is missing, it computes √(c² − other leg²). Every step is shown, including the squares, the sum or difference, and the square root, so you can follow the arithmetic and check it by hand.
The tool also guards against impossible inputs. If you ask for a leg but give a hypotenuse shorter than the other leg, the square root would be of a negative number, which has no real answer. The tool tells you the hypotenuse must be longer than the leg, instead of printing a nonsense result. That guard catches the most common data-entry slip: typing the hypotenuse value into a leg box.
A Worked Example: The Ladder
A ladder leans against a wall. Its foot is 3 metres from the wall and its top reaches 4 metres up the wall. How long is the ladder? The wall and the ground meet at a right angle, so the ladder is the hypotenuse: c² = 3² + 4² = 9 + 16 = 25, so c = 5 metres. Enter a = 3, b = 4 with c missing, and the tool shows exactly this chain.
The same pattern solves countless practical problems: the diagonal of a rectangular garden bed, the straight-line distance across a field when you know the east and north offsets, the length of a ramp, or the distance from a corner to the centre of a room. Whenever two sides meet at a right angle and you need the third side, this theorem is the tool.
Pythagorean Triples
Some right triangles have all three sides as whole numbers. The best known are 3-4-5, 5-12-13 and 8-15-17, and every multiple of a triple is also a triple: 6-8-10 works because it is 3-4-5 doubled. Builders have used the 3-4-5 triangle for centuries to check square corners: measure 3 units along one edge, 4 along the other, and if the diagonal is 5, the corner is square. Try 6-8-10 in the tool and watch the same arithmetic land on a whole number.
Not every triangle gives a whole answer. With legs 1 and 1, the hypotenuse is √2, about 1.4142, a number that never ends and never repeats. The ancient Greeks found this shocking, because it proved that not every length can be written as a simple fraction. The tool prints four decimal places, which is plenty for building, drawing and homework.
Common Mistakes
The most common mistake is applying the theorem to a triangle that has no right angle. The equation a² + b² = c² is only true when the angle between a and b is exactly 90 degrees. For other triangles you need the law of cosines, which is the general version of this theorem. The second mistake is forgetting to take the square root at the end, and answering 25 when the side is 5.
A third mistake is mixing up which side is the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side. If your computed hypotenuse comes out shorter than a leg, you have swapped the labels. Check the labels first; the arithmetic is usually fine.
Try It Yourself
Start with 3 and 4, then try 5 and 12, then try a leg problem: hypotenuse 13, one leg 5, find the other leg, which is 12. Then try 1 and 1 and notice the answer is not a whole number. That spread — whole answers, then an irrational one — is the whole story of the theorem in four tries. Use the tool as often as you like; every calculation runs in your browser, and nothing you type is sent anywhere.
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