Pythagorean Theorem Calculator
Calculate any side of a right triangle using the Pythagorean theorem. Solve for hypotenuse or either leg.
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How to use this calculator
In a right triangle, the square of the hypotenuse (c) equals the sum of squares of the two legs (a and b). Leave one field at 0 to solve for it.
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Enter two known sides of the right triangle.
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Set the unknown side to 0.
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The calculator solves for the missing side and shows all angles and area.
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The hypotenuse is always the longest side, opposite the right angle.
Frequently asked questions
What are Pythagorean triples?
Pythagorean triples are integer solutions: 3-4-5, 5-12-13, 8-15-17, 7-24-25. The family (3k, 4k, 5k) generates infinitely many — e.g. 6-8-10, 9-12-15. These appear in construction for checking right angles: a "3-4-5 check."
How is the Pythagorean theorem used in real life?
Construction (checking if corners are square), navigation (straight-line distance from lat/lon changes), screen diagonal (15" laptop: if 13.4" wide and 7.5" tall → √(13.4² + 7.5²) ≈ 15.4"), and physics (resultant force of two perpendicular vectors).
Does it work in 3D?
Yes. The 3D distance formula is d = √(a² + b² + c²), which extends the theorem to three dimensions. Used in 3D graphics, GPS (latitude, longitude, and altitude), and physics (magnitude of 3D vectors).
Who proved the Pythagorean theorem?
It is named after Pythagoras (~570–495 BC), but the relationship was known to Babylonian and Indian mathematicians centuries earlier. Euclid's Elements contains a formal proof (Book I, Proposition 47). There are now over 370 known proofs of this theorem.
The Pythagorean theorem explained
Understanding a² + b² = c²
In any right triangle, the area of the square built on the hypotenuse equals the combined area of squares built on the two legs. This geometric interpretation — not just an algebraic formula — is why the theorem has such deep significance in mathematics.
The 3-4-5 trick in construction
To check if a corner is a perfect right angle: measure 3 units along one wall, 4 units along the other. If the diagonal measures exactly 5, the corner is square. This "3-4-5 check" has been used by builders for at least 4,000 years.
Non-Euclidean geometry and the theorem
The Pythagorean theorem holds exactly only on flat (Euclidean) surfaces. On a sphere (like Earth), triangles have angle sums greater than 180° and the theorem gives only an approximation. This is why GPS calculations use spherical trigonometry rather than the flat-earth Pythagorean formula.
Pythagorean theorem: how it works
Built on well-established mathematical principles, this tool delivers accurate results for students, researchers, and professionals. Enter your values and get instant clarity without specialist software or manual arithmetic.
Who uses this tool?
Teachers, students, engineers, and data analysts use it to verify calculations, check homework, and solve problems faster. It is intuitive enough for beginners while comprehensive enough for professional use.
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