About Pythagorean Theorem Calculator
This Pythagorean theorem calculator is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded. Give it two sides of a right triangle and it finds the third with each step written out: the squares, their sum or difference, and the square root.
A fourth mode takes three sides in any order and tests them. It tells you whether they form a right, acute or obtuse triangle by comparing the square of the longest side with the sum of the other two squares, and it recognises Pythagorean triples such as 3, 4, 5 and multiples like 6, 8, 10, naming the primitive triple and the Euclid parameters behind it.
The diagram draws a square on each side of the triangle and labels each with its area, so you can see that the two small squares add up to the large one. Choose from eight length units for the labels.
How to use Pythagorean Theorem Calculator
- Choose what to find: the hypotenuse c, leg a, leg b, or check three sides.
- Type the known sides. Only the boxes you need are shown.
- Pick a length unit if you want it printed in the answer.
- Read the answer, the area, the perimeter and the two acute angles.
- Open the step-by-step working to see the squares, the subtraction or addition and the square root.
- In check mode, read the verdict and the triple note, for example whether the sides are a primitive triple.
Features
- Find the hypotenuse, either leg, or test three sides.
- Step-by-step working with KaTeX formulas.
- Pythagorean triple detection, with the primitive triple and Euclid’s m and n.
- Right, acute or obtuse verdict from the converse of the theorem.
- A diagram with a labelled square on each side.
- Area, perimeter and both acute angles of the triangle.
- Clear errors, for example when the hypotenuse is not the longest side.
The Pythagorean theorem
In a right triangle the square of the hypotenuse c, the side opposite the right angle, equals the sum of the squares of the two legs a and b. Rearranged, each leg is the square root of c squared minus the other leg squared.
The converse also holds: if the squares of two sides add up to the square of the third, the triangle is right-angled. If the sum is larger the triangle is acute, and if it is smaller the triangle is obtuse.
a² + b² = c²
c = √(a² + b²)
a = √(c² − b²)
b = √(c² − a²)
Primitive triples: a = m² − n², b = 2mn, c = m² + n²Worked examples
Legs a = 5, b = 12 gives c = 13
5² + 12² = 25 + 144 = 169, and the square root of 169 is 13.
Hypotenuse c = 13, leg b = 12 gives a = 5
13² − 12² = 169 − 144 = 25, so a = 5.
Legs a = 1, b = 1 (the diagonal of a unit square) gives c = 1.414214
The square root of 2, an irrational number.
Check 20, 21, 29 gives Right triangle, primitive triple
20² + 21² = 400 + 441 = 841 = 29².
Pythagorean triples with a hypotenuse up to 50
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 6 | 8 | 10 |
| 5 | 12 | 13 |
| 9 | 12 | 15 |
| 8 | 15 | 17 |
| 12 | 16 | 20 |
| 7 | 24 | 25 |
| 15 | 20 | 25 |
| 10 | 24 | 26 |
| 20 | 21 | 29 |
| 18 | 24 | 30 |
| 16 | 30 | 34 |
| 21 | 28 | 35 |
| 12 | 35 | 37 |
| 15 | 36 | 39 |
| 24 | 32 | 40 |
| 9 | 40 | 41 |
| 27 | 36 | 45 |
| 14 | 48 | 50 |
| 30 | 40 | 50 |
Good to know
Where the theorem applies
The theorem holds only for right triangles in flat (Euclidean) geometry. For other triangles use the law of cosines, which reduces to the Pythagorean theorem when the angle is 90°. The triangle calculator on this site solves general triangles.
Privacy: how it works
- All the arithmetic runs in your browser tab. The sides you enter are not sent to Ulkaa or to anyone else, and there is nothing to sign in to.
Tips
- The hypotenuse is always the longest side. If you enter a leg longer than c, the calculator refuses and says so.
- To test a right angle on site, measure 3 units along one wall and 4 along the other. The diagonal should be 5.
- Keep all sides in the same unit. The calculator does not convert between units for you.
- A triple needs whole numbers. 1.5, 2, 2.5 satisfy the theorem but are not counted as an integer triple.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your values are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do you find the hypotenuse?
Square both legs, add the squares and take the square root: c = √(a² + b²). For legs 5 and 12 that is √169 = 13.
How do you find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse and take the square root: a = √(c² − b²).
What is a Pythagorean triple?
Three positive whole numbers a, b and c with a² + b² = c², such as 3, 4, 5 or 5, 12, 13. A primitive triple has no common factor, and every other triple is a multiple of one.
How can I tell if a triangle is right, acute or obtuse from its sides?
Order the sides so c is longest and compare c² with a² + b². Equal means right, c² smaller means acute, and c² larger means obtuse.
Can the sides be decimals?
Yes. The theorem works for any positive real numbers. Only the triple check needs whole numbers.
Does the theorem work in 3D?
It extends: the diagonal of a box with edges a, b, c is √(a² + b² + c²). Apply the theorem twice, first on the base and then with the height.
Popular Pythagorean Theorem Calculator pages
- Hypotenuse CalculatorEnter the two legs and get the hypotenuse.
- Missing Leg CalculatorEnter one leg and the hypotenuse to find the other leg.
- Find Leg b CalculatorEnter leg a and the hypotenuse to solve for leg b.
- 3-4-5 Triangle CalculatorSee why 3, 4 and 5 make a right angle and how to scale it.
- Pythagorean Triples CalculatorEnter three whole numbers to test them and see the primitive triple and Euclid’s m and n.
- Right Triangle CheckerEnter three sides in any order and see whether the triangle is right, acute or obtuse.
- Converse of the Pythagorean Theorem CalculatorEnter three sides and apply the converse to classify the triangle.