Free Pythagorean theorem calculator
Solve right triangles using the Pythagorean theorem — enter any two sides to find the third: a² + b² = c², updated live, as you type.
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a² + b² = c². Source: Euclid’s Elements.
Results are estimates. Consult a professional.
How the Pythagorean theorem calculator works
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. The calculator solves for any one side when the other two are known.
Worked example: legs 6 and 8
A right triangle has legs of length 6 and 8. This is a scaled version of the classic 3-4-5 triple (×2).
Common Pythagorean triples
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Memorising the primitive triples (where GCF = 1) lets you recognise right triangles instantly.
| a | b | c | Type |
|---|---|---|---|
| 3 | 4 | 5 | Primitive |
| 6 | 8 | 10 | 3-4-5 × 2 |
| 5 | 12 | 13 | Primitive |
| 8 | 15 | 17 | Primitive |
| 7 | 24 | 25 | Primitive |
| 9 | 40 | 41 | Primitive |
| 10 | 24 | 26 | 5-12-13 × 2 |
| 20 | 21 | 29 | Primitive |
| 12 | 35 | 37 | Primitive |
| 11 | 60 | 61 | Primitive |
Source: standard list of Pythagorean triples; verify with a² + b² = c²
Tips for applying the Pythagorean theorem
The theorem appears in architecture, navigation, engineering, and video games. These tips help you apply it without errors.
- Identify the hypotenuse first — it is always opposite the 90° angle and is the longest side; mislabelling it is the most common mistake.
- Recognise scaled triples — if two sides have a 3:4 ratio (or 5:12, or 8:15), the third side completes a triple; no calculation needed.
- Use it in 3D with two steps — to find the space diagonal of a box (l × w × h), first find the floor diagonal d = √(l²+w²), then apply again: space diagonal = √(d²+h²).
- Converse test for right angles — if a² + b² = c² for three sides, the triangle must contain a right angle; useful for checking whether a corner is truly square in construction.
- Distance formula is Pythagoras in disguise — the distance between points (x₁,y₁) and (x₂,y₂) is √((x₂−x₁)² + (y₂−y₁)²), which is just the hypotenuse of a right triangle.
Accuracy and limitations
The Pythagorean theorem applies only to right triangles in flat (Euclidean) geometry. On curved surfaces — such as Earth's surface in navigation or on a globe — the theorem does not hold; use spherical trigonometry instead. Results are computed using IEEE 754 double-precision square root, which is accurate to approximately 15 significant digits. For irrational results (e.g. √2, √5) the calculator returns a decimal approximation; for exact symbolic answers use a computer algebra system.
Key terms
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