Hypotenuse Calculator
Solve any side of a right triangle with the Pythagorean theorem: a² + b² = c². Pick a mode, enter two sides, and get the third — with a live triangle diagram, unit conversions, and a Pythagorean triple checker.
Triangle diagram
Pythagorean triple checker
Enter three sides (largest last) to check whether they form a Pythagorean triple.
Worked examples
Tap an example to load it into the calculator.
Not what you need? This page solves for the hypotenuse (or a missing leg) of a right triangle using the Pythagorean theorem. If you want the area of a triangle instead, use our triangle area calculator. For the volume of a right-triangle-based prism, see our right triangle volume calculator.
The Pythagorean theorem, explained
The Pythagorean theorem describes the relationship between the three sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the two legs.
a² + b² = c²
| Symbol | Meaning | How to find it |
|---|---|---|
| a | First leg (one side forming the right angle) | a = √(c² − b²) |
| b | Second leg (the other side forming the right angle) | b = √(c² − a²) |
| c | Hypotenuse (longest side, opposite the 90° angle) | c = √(a² + b²) |
Why it works: if you draw a square on each side of the triangle, the area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. For a 3-4-5 triangle: 3² + 4² = 9 + 16 = 25 = 5².
Pythagorean triples reference table
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfies a² + b² = c². Primitive triples cannot be reduced; scaled triples are multiples of a primitive triple.
| a | b | c | Type | Check |
|---|---|---|---|---|
| 3 | 4 | 5 | Primitive | 9 + 16 = 25 |
| 5 | 12 | 13 | Primitive | 25 + 144 = 169 |
| 8 | 15 | 17 | Primitive | 64 + 225 = 289 |
| 7 | 24 | 25 | Primitive | 49 + 576 = 625 |
| 20 | 21 | 29 | Primitive | 400 + 441 = 841 |
| 9 | 40 | 41 | Primitive | 81 + 1600 = 1681 |
| 12 | 35 | 37 | Primitive | 144 + 1225 = 1369 |
| 11 | 60 | 61 | Primitive | 121 + 3600 = 3721 |
| 28 | 45 | 53 | Primitive | 784 + 2025 = 2809 |
| 33 | 56 | 65 | Primitive | 1089 + 3136 = 4225 |
| 13 | 84 | 85 | Primitive | 169 + 7056 = 7225 |
| 48 | 55 | 73 | Primitive | 2304 + 3025 = 5329 |
| 39 | 80 | 89 | Primitive | 1521 + 6400 = 7921 |
| 65 | 72 | 97 | Primitive | 4225 + 5184 = 9409 |
| 15 | 112 | 113 | Primitive | 225 + 12544 = 12769 |
| 6 | 8 | 10 | Scaled (3-4-5 × 2) | 36 + 64 = 100 |
| 9 | 12 | 15 | Scaled (3-4-5 × 3) | 81 + 144 = 225 |
| 10 | 24 | 26 | Scaled (5-12-13 × 2) | 100 + 576 = 676 |
| 14 | 48 | 50 | Scaled (7-24-25 × 2) | 196 + 2304 = 2500 |
| 16 | 30 | 34 | Scaled (8-15-17 × 2) | 256 + 900 = 1156 |
Pythagorean triple generator
Enter a maximum hypotenuse to list every primitive triple up to that value, generated with Euclid's formula (a = m² − n², b = 2mn, c = m² + n²).
Real-world applications
| Field | How the theorem is used |
|---|---|
| Construction | The 3-4-5 rule: builders measure 3 ft along one wall and 4 ft along the adjacent wall; if the diagonal is exactly 5 ft, the corner is square. |
| Navigation | GPS receivers use trilateration, which relies on right-triangle geometry to compute distances from satellites. |
| Screen sizes | A 55-inch TV is measured diagonally: a 48.5″ × 27.3″ screen gives √(48.5² + 27.3²) ≈ 55.7″. |
| Ladders | A ladder 5 m long with its base 3 m from the wall reaches √(25 − 9) = 4 m up the wall. |
| Sports | Baseball: the throw from home plate to second base crosses a 90 ft × 90 ft square, so the distance is √(90² + 90²) ≈ 127.3 ft. |
| Computer graphics | Distance between two pixels (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²). |
Common mistakes to avoid
1. Applying it to non-right triangles
The theorem only works when one angle is exactly 90°. For other triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C).
2. Forgetting to take the square root
a² + b² gives c², not c. With legs 3 and 4: 9 + 16 = 25, so c = √25 = 5, not 25.
3. Mixing up which side is the hypotenuse
The hypotenuse is always the longest side and always opposite the right angle. In the formula a² + b² = c², c must be the largest value.
4. Adding instead of subtracting when solving for a leg
To find a leg, subtract: b = √(c² − a²). Adding would give a value larger than the hypotenuse, which is impossible.
Related calculators
Frequently asked questions
What is the hypotenuse of a right triangle?
The hypotenuse is the longest side of a right triangle — the side opposite the 90° angle. By the Pythagorean theorem, its length is c = √(a² + b²), where a and b are the two legs.
How do I find a missing leg of a right triangle?
Rearrange the theorem: if you know the hypotenuse c and one leg a, the other leg is b = √(c² − a²). The hypotenuse must be longer than the known leg, otherwise there is no valid right triangle.
What are the most common Pythagorean triples?
Common Pythagorean triples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple — for example (6, 8, 10) is just (3, 4, 5) scaled by 2.
Can I use decimals and different length units?
Yes. You can enter decimals, and you can choose mm, cm, m, km, in, ft, yd, or mi as your input unit. The calculator shows the answer in your unit plus automatic conversions to all other supported units.
How do you find the hypotenuse with the Pythagorean theorem?
Square both legs, add them, then take the square root: c = √(a² + b²). For legs 3 and 4: √(9 + 16) = √25 = 5.
What is the 3-4-5 rule?
The 3-4-5 rule is the simplest Pythagorean triple: 3² + 4² = 9 + 16 = 25 = 5². Builders use it to check square corners: measure 3 units along one wall and 4 along the other; if the diagonal is exactly 5, the corner is a true right angle.
Can the Pythagorean theorem find angles?
Indirectly. Once you know all three sides, each acute angle comes from the inverse tangent: angle = arctan(opposite/adjacent). For a 3-4-5 triangle the acute angles are arctan(3/4) ≈ 36.87° and 53.13°. This calculator shows both angles automatically.
Does the Pythagorean theorem work for non-right triangles?
No. It only applies when one angle is exactly 90°. For any other triangle, use the law of cosines: c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when C = 90°.
What is a Pythagorean triple?
A Pythagorean triple is three positive integers (a, b, c) satisfying a² + b² = c², such as (3, 4, 5) or (5, 12, 13). A triple is called primitive when the three numbers share no common factor. Use the triple checker above to test any three sides.
How do you generate Pythagorean triples?
Use Euclid’s formula: pick integers m > n with no common factor and opposite parity, then a = m² − n², b = 2mn, c = m² + n². For m = 2, n = 1: a = 3, b = 4, c = 5. The triple generator above automates this.
What is the converse of the Pythagorean theorem?
The converse states that if a² + b² = c² for a triangle’s three sides, then the triangle must contain a right angle opposite side c. This is how inspectors verify square corners without measuring angles.
Who discovered the Pythagorean theorem?
It is credited to the Greek mathematician Pythagoras (6th century BC), though Babylonian tablets such as Plimpton 322 show the relationship was known and used more than a thousand years earlier.
What are real-world uses of the Pythagorean theorem?
Construction (squaring corners with the 3-4-5 rule), navigation and GPS (distance calculations), screen-size measurement (diagonal), ladder safety, sports-field layouts, and computer graphics (pixel distances) all rely on it.
Can the sides be decimals or fractions?
Yes. The theorem works for any positive real numbers. For legs 1.5 and 2: c = √(2.25 + 4) = √6.25 = 2.5, which is the (3, 4, 5) triple scaled by 0.5.
Is there a 3D version of the Pythagorean theorem?
Yes. The space diagonal of a rectangular box is d = √(a² + b² + c²). For a 1 × 2 × 2 box: d = √(1 + 4 + 4) = √9 = 3. Apply the 2D theorem twice: first on the base, then on the vertical cross-section.
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