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About this tool
Find the hypotenuse or a missing leg with a² + b² = c², get the simplified radical and decimal, see steps, and check or generate Pythagorean triples.
This Pythagorean Theorem Calculator solves a right triangle from any two sides. Choose whether you want the hypotenuse c or a leg, enter the two known lengths, and it applies c² = a² + b² (or a² = c² − b²). Besides the decimal, it keeps the arithmetic exact: the inputs are scaled to whole numbers and the square factors are pulled out of the root, so 5 and 7 give √74 and 5 and 5 give 5√2. A labelled diagram, the area, perimeter, both acute angles and the steps are shown. Separate tabs check whether three sides form a right triangle and generate Pythagorean triples. It all runs in your browser.
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In Solve a² + b² = c², choose Hypotenuse c, Leg a or Leg b as the side to find.
Enter the two known lengths, for example legs 5 and 7, and pick a Unit such as cm and the Decimal places.
Read the simplified radical (√74) and decimal (≈ 8.6023), the right-triangle diagram, area, angles and Steps, then press Copy.
Use Check sides to test three lengths like 6, 8, 10, or Triples to list Pythagorean triples up to a largest hypotenuse and download them as CSV.
Results are given in simplified radical form, such as 2√3, alongside a decimal rounded to the places you choose, so you can match either style of answer.
The formula, the squared values, the sum or difference and the square-root simplification are listed in order with your numbers.
Test any three lengths with the converse of the theorem, see if they are a primitive or scaled Pythagorean triple, and list every triple up to a chosen hypotenuse.
Pick mm, cm, m, km, in, ft, yd or mi; lengths and the area in square units are labelled, and the solved side is highlighted on a scaled drawing.
| a | b | 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 |
| 12 | 35 | 37 | 144 + 1225 = 1369 |
| 9 | 40 | 41 | 81 + 1600 = 1681 |
| 28 | 45 | 53 | 784 + 2025 = 2809 |
| 11 | 60 | 61 | 121 + 3600 = 3721 |
| 16 | 63 | 65 | 256 + 3969 = 4225 |
| 33 | 56 | 65 | 1089 + 3136 = 4225 |
| 48 | 55 | 73 | 2304 + 3025 = 5329 |
| 13 | 84 | 85 | 169 + 7056 = 7225 |
| 36 | 77 | 85 | 1296 + 5929 = 7225 |
| 39 | 80 | 89 | 1521 + 6400 = 7921 |
| 65 | 72 | 97 | 4225 + 5184 = 9409 |
Multiply any row by a whole number to get a non-primitive triple, e.g. 3-4-5 × 2 = 6-8-10.
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 c is the side opposite the right angle and is always the longest. The theorem only applies to right triangles; for other triangles the law of cosines adds a correction term.
Square both legs, add them, and take the square root: c = √(a² + b²). For legs 6 and 8, c = √(36 + 64) = √100 = 10. If the sum is not a perfect square, the answer is irrational; for legs 1 and 2 it is √5, about 2.2361.
Rearrange the formula: a = √(c² − b²). Square the hypotenuse, subtract the square of the known leg, and take the square root. For a hypotenuse of 13 and a leg of 12, a = √(169 − 144) = √25 = 5. The hypotenuse must be longer than the leg, or there is no solution.
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c², such as 3, 4, 5 or 5, 12, 13. A triple is primitive when the three numbers share no common factor. Multiplying a triple by any whole number gives another triple, so 6, 8, 10 is 3, 4, 5 doubled.
Euclid's formula produces every primitive triple: pick whole numbers m > n that share no factor and are not both odd, then a = m² − n², b = 2mn and c = m² + n². With m = 2 and n = 1 you get 3, 4, 5; with m = 3 and n = 2 you get 5, 12, 13.
Use the converse of the theorem: if the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. If the sum is greater, the triangle is acute; if smaller, it is obtuse. The Check sides tab does this comparison and also flags whole-number triples.
Factor the number under the square root and take out any perfect-square factor. For example √72 = √(36 × 2) = 6√2, and √50 = √(25 × 2) = 5√2. If no square factor other than 1 divides the number, as with √74, the radical is already in simplest form.
Builders mark 3 units along one line and 4 along the other; if the diagonal between the marks measures exactly 5, the corner is a right angle. Any multiple works, such as 6-8-10 or 9-12-15 feet, and larger multiples give a more accurate check over a long wall.
Yes, by applying it twice. The space diagonal of a box with sides x, y and z is √(x² + y² + z²): first find the diagonal of the base, √(x² + y²), then use it as one leg with the height z as the other. You can do the two steps here one after the other.
Yes. Decimals like 1.5 and 2 work and give exact answers when possible, here 2.5. Both inputs must use the same unit; the result is in that unit and the area in that unit squared. Convert first if your measurements are mixed, such as feet and inches.
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