Quick Answer
Quick Answer: To find the hypotenuse, square both legs of the right triangle, add the squares, and take the square root: c = √(a² + b²). For legs 6 and 8, that is 6² + 8² = 36 + 64 = 100, so the hypotenuse is √100 = 10. Calcrio’s free pythagorean theorem calculator shows every step.
What Is the Hypotenuse?
The hypotenuse is the longest side of a right triangle, and it sits opposite the 90° angle. The other two sides are called the legs. If you know the two legs, the hypotenuse is fixed — there is exactly one length that completes the triangle, and the Pythagorean theorem finds it.
Every hypotenuse question is the same question in disguise: the square of the hypotenuse equals the sum of the squares of the legs. Once that clicks, finding the hypotenuse is three small steps: square, add, square root. The most common error is skipping one of those steps — adding the legs instead of their squares, or forgetting the square root at the end.
The hypotenuse shows up far beyond textbook triangles. It is the diagonal of a rectangle, the length of a ladder leaning on a wall, and the distance between two points on a map. Learning to find it by hand first, then checking with a tool, is the fastest way to make the method stick.
How to Find the Hypotenuse Step by Step
Step 1: Identify the Two Legs
Look at the triangle and name its two legs — the sides that form the right angle. In a triangle with legs 6 and 8, a = 6 and b = 8. The hypotenuse is the side you do not know yet, opposite the right angle. Never use the hypotenuse as one of the legs: it is always the longest side.
Step 2: Square Both Legs
Square each leg separately. For legs 6 and 8: 6² = 36 and 8² = 64. Write both squares down — keeping them separate makes the addition easy to check.
Step 3: Add the Squares
Add the two squares: 36 + 64 = 100. This number is c², the square of the hypotenuse — not the hypotenuse itself. That distinction is the whole trick of the method.
Step 4: Take the Square Root
Take the square root to undo the square: √100 = 10. So the hypotenuse is 10. Check it backwards: 10² = 100, and 36 + 64 = 100 — the two sides match exactly. Or verify instantly with Calcrio’s pythagorean theorem calculator, which shows the squares, the sum, and the square root.
Worked Examples
| Legs | Squares | Sum | Hypotenuse |
|---|---|---|---|
| 3 and 4 | 9 + 16 | 25 | 5 |
| 6 and 8 | 36 + 64 | 100 | 10 |
| 5 and 12 | 25 + 144 | 169 | 13 |
| 8 and 15 | 64 + 225 | 289 | 17 |
| 7 and 24 | 49 + 576 | 625 | 25 |
| 9 and 12 | 81 + 144 | 225 | 15 |
| 1 and 1 | 1 + 1 | 2 | ≈ 1.4142 |
Example: Legs 6 and 8 — the Hypotenuse Is 10
Square the legs: 6² = 36 and 8² = 64. Add: 36 + 64 = 100. Square root: √100 = 10. This is the most useful hypotenuse to know by heart, because 6-8-10 is the 3-4-5 triple doubled — and it is the ladder problem, the rectangle problem, and half of all textbook problems in one example.
Example: Legs 5 and 12 — the Hypotenuse Is 13
Square the legs: 5² = 25 and 12² = 144. Add: 25 + 144 = 169. Square root: √169 = 13. Check: 13² = 169. The 5-12-13 triple is the second triple worth memorizing, after 3-4-5.
Example: Legs 1 and 1 — the Hypotenuse Is √2
Square the legs: 1² + 1² = 2. The hypotenuse is √2 ≈ 1.4142. Not every hypotenuse is a whole number. When the sum of the squares is not a perfect square, the answer is irrational, and four decimal places (1.4142) is the practical answer. This is the diagonal of a unit square — a shape that appears in geometry, design, and paper sizes.
What Are Pythagorean Triples?
A Pythagorean triple is a set of three whole numbers that satisfies a² + b² = c² exactly. The most famous is 3-4-5: 9 + 16 = 25. Others worth knowing are 5-12-13 (25 + 144 = 169), 8-15-17 (64 + 225 = 289), 7-24-25 (49 + 576 = 625), and 20-21-29 (400 + 441 = 841).
Multiples of a triple are triples too. Doubling 3-4-5 gives 6-8-10 (36 + 64 = 100). Tripling it gives 9-12-15 (81 + 144 = 225). Doubling 5-12-13 gives 10-24-26. Recognizing a triple saves the whole calculation: if the legs are 6 and 8, the hypotenuse is 10 without reaching for a calculator.
| Triple | Check | Hypotenuse |
|---|---|---|
| 3-4-5 | 9 + 16 = 25 | 5 |
| 5-12-13 | 25 + 144 = 169 | 13 |
| 8-15-17 | 64 + 225 = 289 | 17 |
| 7-24-25 | 49 + 576 = 625 | 25 |
| 9-12-15 | 81 + 144 = 225 | 15 |
| 20-21-29 | 400 + 441 = 841 | 29 |
How Do You Find the Hypotenuse Without a Calculator?
The method is the same — square, add, square root — but the arithmetic is easier when you know the triples and the perfect squares. The squares up to 25² are worth knowing: 3² = 9, 4² = 16, 5² = 25, 6² = 36, 8² = 64, 12² = 144, 13² = 169, 15² = 225, 17² = 289, 24² = 576, 25² = 625.
For legs 9 and 12, the squares are 81 and 144. Add them in your head: 81 + 144 = 225. Is 225 a perfect square? 15² = 225, so the hypotenuse is 15. The check is built in: if the sum is a perfect square you recognize, the answer is its root. If it is not — like 1² + 1² = 2 — the answer is a square root, and a calculator gives the decimal. Calcrio’s how to find the hypotenuse tool gives the exact value and the decimal side by side.
What If You Know the Hypotenuse and One Leg?
Then you are finding a leg, not the hypotenuse, and the method flips: subtract instead of add. The leg is b = √(c² − a²). For a hypotenuse of 13 and a leg of 5: b = √(169 − 25) = √144 = 12. For a hypotenuse of 17 and a leg of 8: b = √(289 − 64) = √225 = 15. For a hypotenuse of 25 and a leg of 7: b = √(625 − 49) = √576 = 24.
The rule for which operation to use is simple: finding the longest side means adding squares; finding a shorter side means subtracting. If you ever get a negative number under the square root, you have mixed them up — the hypotenuse must be the longest side.
Where Is the Hypotenuse Used in Real Life?
The hypotenuse is the diagonal of a rectangle, so any screen, page, or room measured corner to corner uses it. A 3 × 4 rectangle has diagonal √(9 + 16) = 5. A 5 × 12 rectangle has diagonal √(25 + 144) = 13.
It is the ladder problem: a 10-foot ladder with its foot 6 feet from a wall reaches √(100 − 36) = 8 feet up the wall. Builders use the same idea in reverse — the 3-4-5 method squares a corner by marking 3 feet along one edge, 4 feet along the other, and checking the diagonal is exactly 5 feet.
And it is the distance between two points: the points (3, 4) and (7, 1) are 4 units apart horizontally and 3 units apart vertically, so the straight-line distance is √(16 + 9) = 5. That is the distance formula, and it is the Pythagorean theorem wearing different clothes.
Common Mistakes When Finding the Hypotenuse
1. Adding the Legs Instead of Their Squares
For legs 6 and 8, a + b = 14 — but the hypotenuse is 10. Square first, then add, then take the square root. Adding the sides directly always overestimates the answer.
2. Forgetting the Square Root
36 + 64 = 100 is c², not c. The hypotenuse is √100 = 10, not 100. If your answer is bigger than the sum of the legs, you forgot the square root.
3. Using the Theorem on a Triangle That Is Not Right-Angled
The theorem only applies when one angle is exactly 90°. For any other triangle it gives the wrong answer. Check the triangle is right-angled first.
4. Calling a Leg the Hypotenuse
The hypotenuse is always the longest side and always opposite the right angle. If the side you are solving for is not the longest, you are finding a leg — subtract, don’t add.
5. Rounding Too Early
Keep the full squares until the last step. Rounding 1.4142 to 1.41 before squaring it back gives 1.9881, not 2. Square first, add, then round only the final answer.
Frequently Asked Questions
How do you find the hypotenuse of a right triangle?
Square both legs, add the squares, and take the square root: c = √(a² + b²). For legs 6 and 8: 6² + 8² = 36 + 64 = 100, and √100 = 10.
What is the formula for the hypotenuse?
c = √(a² + b²), where a and b are the legs and c is the hypotenuse. It is the Pythagorean theorem, a² + b² = c², rearranged to solve for c.
How do you find the hypotenuse with legs 3 and 4?
3² + 4² = 9 + 16 = 25, and √25 = 5. The hypotenuse is 5. This is the 3-4-5 Pythagorean triple, the most famous in mathematics.
Can the hypotenuse be shorter than a leg?
No. The hypotenuse is always the longest side of a right triangle, because it is the square root of the sum of the squares of the legs — and that sum is always larger than either leg’s square alone.
How do you find the hypotenuse without a calculator?
Square both legs by hand, add them, and check whether the sum is a perfect square you recognize. For legs 9 and 12: 81 + 144 = 225 = 15², so the hypotenuse is 15. If the sum is not a perfect square, the answer is a square root like √2 ≈ 1.4142.
What is the hypotenuse of a 5-12-13 triangle?
13, because 5² + 12² = 25 + 144 = 169 = 13². The 5-12-13 triple is the second triple worth memorizing after 3-4-5.