Find any missing side of a right triangle in seconds. Enter the two sides you know and get the hypotenuse, a missing leg, or a right-triangle check — with every step shown.
Method: a² + b² = c² for a right triangle.
Find a hypotenuse, a missing leg, or check whether three sides make a right triangle.
Enter the three sides in any order — the longest side is taken as the hypotenuse.
This tool runs in your browser and does not send your calculations anywhere. Whole-number answers are exact; other answers are rounded to 4 decimal places.
Quick Answer: The Pythagorean theorem says that in a right triangle, a² + b² = c², where c is the hypotenuse. Calcrio's free Pythagorean theorem calculator finds the hypotenuse (6² + 8² = 10²), finds a missing leg (√(13² − 5²) = 12), or checks whether three sides make a right triangle, with every step shown.
Choose a mode, enter the sides you know, and select Calculate.
The tool runs in your browser, is free, and needs no sign-up. Whole-number answers are exact; other answers are rounded to 4 decimal places.
A right triangle has one angle of 90°. The side opposite the right angle is the hypotenuse — always the longest side. The other two sides are the legs, and they form the right angle.
The theorem is named after Pythagoras (c. 570–495 BCE), although the relationship was known earlier in Babylonian and Egyptian mathematics. It states that the square of the hypotenuse equals the sum of the squares of the legs.
a² + b² = c²
For a 3-4-5 triangle, the square on the hypotenuse is 25, and the squares on the legs add to 9 + 16 = 25. The two areas are equal.
| Leg a | Leg b | Hypotenuse c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 3² + 4² = 9 + 16 = 25 = 5² |
| 6 | 8 | 10 | 6² + 8² = 36 + 64 = 100 = 10² |
| 5 | 12 | 13 | 5² + 12² = 25 + 144 = 169 = 13² |
| 8 | 15 | 17 | 8² + 15² = 64 + 225 = 289 = 17² |
| 7 | 24 | 25 | 7² + 24² = 49 + 576 = 625 = 25² |
| 9 | 12 | 15 | 9² + 12² = 81 + 144 = 225 = 15² |
| 20 | 21 | 29 | 20² + 21² = 400 + 441 = 841 = 29² |
| 1 | 1 | √2 | 1² + 1² = 1 + 1 = 2 = (√2)², hypotenuse ≈ 1.4142 |
Legs a = 6 and b = 8. c = √(6² + 8²) = √(36 + 64) = √100 = 10. Check: 6² + 8² = 36 + 64 = 100 = 10².
Hypotenuse c = 13 and leg a = 5. b = √(13² − 5²) = √(169 − 25) = √144 = 12. Check: 5² + 12² = 25 + 144 = 169 = 13².
A 10-foot ladder stands with its foot 6 feet from a wall. The height it reaches is b = √(10² − 6²) = √(100 − 36) = √64 = 8 feet. Check: 6² + 8² = 36 + 64 = 100 = 10². This is the 6-8-10 triangle, which is the 3-4-5 triple doubled.
Use c = √(a² + b²). For a = 5 and b = 12: c = √(25 + 144) = √169 = 13. For a = 9 and b = 12: c = √(81 + 144) = √225 = 15. For a = 1 and b = 1: c = √(1 + 1) = √2 ≈ 1.4142.
Use b = √(c² − a²). For c = 13 and a = 5: b = √(169 − 25) = √144 = 12. For c = 17 and a = 8: b = √(289 − 64) = √225 = 15 (check: 8² + 15² = 64 + 225 = 289 = 17²). For c = 25 and a = 7: b = √(625 − 49) = √576 = 24 (check: 7² + 24² = 49 + 576 = 625 = 25²).
Test a² + b² = c² with the longest side as c. 3, 4, 5 is right (9 + 16 = 25). 2, 3, 4 is not right (4 + 9 = 13 ≠ 16). 6, 8, 11 is not right (36 + 64 = 100 ≠ 121).
A rectangle diagonal splits it into two right triangles. For a 3 × 4 rectangle, the diagonal is √(3² + 4²) = √(9 + 16) = √25 = 5. For a 5 × 12 rectangle, the diagonal is √(25 + 144) = √169 = 13.
The distance formula is the Pythagorean theorem in disguise. From (3, 4) to (7, 1): dx = 4 and dy = −3, so the distance is √(4² + (−3)²) = √(16 + 9) = √25 = 5.
| Question | Answer |
|---|---|
| What is the Pythagorean theorem? | In a right triangle, a² + b² = c², where c is the hypotenuse and a and b are the legs. |
| What is the hypotenuse? | The longest side, opposite the 90° angle. For legs 6 and 8 it is 10. |
| What is a Pythagorean triple? | Three whole numbers satisfying a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17. |
| How do you find the hypotenuse? | c = √(a² + b²). |
| How do you find a missing leg? | b = √(c² − a²). For c = 13 and a = 5, b = √(169 − 25) = 12. |
| What is 6² + 8²? | 36 + 64 = 100, so the hypotenuse is √100 = 10. |
| How do you check a right triangle? | Test a² + b² = c². For 3, 4, 5: 9 + 16 = 25, so it is right-angled. |
A free tool that finds any missing side of a right triangle using a² + b² = c². It finds the hypotenuse, finds a missing leg, or checks whether three sides make a right triangle, with every step shown.
Use c = √(a² + b²). For legs 6 and 8: 6² + 8² = 36 + 64 = 100, so c = √100 = 10.
Use b = √(c² − a²). For c = 13 and a = 5: b = √(169 − 25) = √144 = 12.
3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-12-15 and 20-21-29. Doubling 3-4-5 gives 6-8-10, and tripling it gives 9-12-15.
Test a² + b² = c² with the longest side as c. For 3, 4, 5: 9 + 16 = 25, so it is right-angled.
Builders use it to square corners. A 10-foot ladder with its foot 6 feet from a wall reaches 8 feet up the wall. It also gives rectangle diagonals (a 3 × 4 rectangle has diagonal 5) and the distance between two points.
Finding the hypotenuse: c = √(a² + b²). For legs 6 and 8: c = √(36 + 64) = √100 = 10.
Finding a leg: b = √(c² − a²). The calculator validates that c > a and c > 0; otherwise it reports that the hypotenuse must be the longest side. For c = 13 and a = 5: b = √(169 − 25) = √144 = 12.
Checking a right triangle: the longest side is taken as c and a² + b² = c² is tested within a small tolerance (1e-9 relative). If it passes, the calculator reports a right triangle and names the triple when the sides m
Rounding: whole-number results are exact; other results are rounded to 4 decimal places (√2 ≈ 1.4142).
These formulas give deterministic mathematical results from the values entered. The page is intended as an educational calculator and verification tool.
Enter the two sides you know above to get the answer with full worked steps — or test the classic 6-8-10 triangle and watch 36 + 64 land exactly on 100.