Pythagorean theorem calculator

The theorem

In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:

a² + b² = c²

c is always the hypotenuse — the side opposite the right angle, and always the longest. Getting that wrong is the single most common error, and it shows up as a hypotenuse shorter than one of the legs, which is geometrically impossible. The calculator rejects that case rather than returning a nonsense root.

Rearranging for a leg

To find a missing leg rather than the hypotenuse, subtract instead of add: a = √(c² − b²). The subtraction is what makes the ordering matter — if you accidentally subtract the larger square from the smaller you get a negative under the root and no real answer.

Pythagorean triples

Some right triangles have three whole-number sides. The small ones are worth memorising because they appear constantly in exams and on building sites:

• 3, 4, 5 — and every multiple: 6-8-10, 9-12-15, 30-40-50
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

The 3-4-5 triple is the basis of the builder's method for checking a square corner: measure 3 units along one wall, 4 along the other, and if the diagonal between those marks is exactly 5, the corner is exactly 90°. Carpenters usually work in feet and use 6-8-10 for better accuracy over a longer run.

Where it gets used

Straight-line distance between two points is Pythagoras in disguise: the distance formula √((x₂−x₁)² + (y₂−y₁)²) is just the theorem applied to the horizontal and vertical gaps. The same idea extends to three dimensions by adding a third squared term, which is how you find the diagonal of a box.

Practical uses are everywhere: the length of a roof rafter from rise and run, whether a sofa fits diagonally through a doorway, ladder safety angles, screen sizes quoted as a diagonal, and the shortest path across a field rather than round two edges.

Only for right triangles

The theorem holds only when one angle is exactly 90°. For any other triangle you need the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras when C = 90° because cos(90°) = 0.

The converse is also true and genuinely useful: if a² + b² = c² holds for three lengths, the triangle they form must be right-angled. That is what makes the 3-4-5 check valid rather than merely suggestive.

Common questions