Pythagorean Theorem Calculator

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, labelled c) equals the sum of the squares of the other two sides (the legs a and b): a² + b² = c². It is one of the oldest and most fundamental results in geometry, attributed to Pythagoras around 500 BCE — though Babylonian tablets show it was known a thousand years earlier.

Rearrange the theorem: a = √(c² − b²) or b = √(c² − a²). The hypotenuse c must be the largest side, so when solving for a leg, the value under the square root must be positive — otherwise no real triangle exists with those measurements.

A Pythagorean triple is three positive integers (a, b, c) satisfying a² + b² = c². Classic examples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any positive multiple of a triple is also a triple (so 6-8-10 and 9-12-15 are scaled versions of 3-4-5). A triple is called primitive if the three numbers share no common factor.

Yes — strictly. For triangles that are not right-angled, use the law of cosines: c² = a² + b² − 2ab·cos(C). The Pythagorean theorem is exactly this formula in the special case where angle C = 90° (cos 90° = 0, so the last term vanishes). For oblique triangles, also see our Triangle Calculator.

Once all three sides are known, use inverse trigonometry: angle A = arcsin(a / c) = arctan(a / b), and similarly for B. The two non-right angles always sum to 90°, since the angle sum of any triangle is 180° and one corner is already 90°. This calculator returns both angles automatically.

Area = (1/2) × a × b — half the product of the two legs. The legs meet at the right angle and so serve directly as base and height, no perpendicular projection required. (For non-right triangles you need Heron's formula or ½·base·height with a measured perpendicular.)

If you're solving for a leg, the hypotenuse must be larger than the other leg — otherwise c² − b² is negative and its square root is not a real number. Check that you've assigned the longest side as c (the hypotenuse). If you accidentally entered c smaller than a or b, swap them.

The Pythagorean theorem appears constantly. Carpenters use 3-4-5 triangles to check whether a corner is truly square. Surveyors use it to compute straight-line distances. The distance formula in coordinate geometry — d = √((x₂−x₁)² + (y₂−y₁)²) — is just Pythagoras. In 3D, the diagonal of a box is √(l² + w² + h²), and in physics, the magnitude of a 2D vector is exactly √(vₓ² + vᵧ²).

Yes — any positive real number works. The calculator accepts decimals directly. For fractions, convert first (e.g. 3/4 = 0.75). Results are reported to 4 decimal places by default. Most "neat" results — like √2 ≈ 1.4142 or √5 ≈ 2.2361 — are irrational and have non-terminating decimal expansions.

Yes. The converse of the Pythagorean theorem states that if a² + b² = c² for the three sides of a triangle, then the triangle is right-angled (with the right angle opposite c). This is what makes the 3-4-5 trick work for builders — measure three sides and check the equation; if it holds, your corner is square to a tolerance of your measurement.

The result was almost certainly known to Babylonian and Indian mathematicians long before Pythagoras (c. 570–495 BCE). A Babylonian clay tablet (Plimpton 322, c. 1800 BCE) lists Pythagorean triples. Pythagoras (or his school) is credited with the first known proof of the result in the western tradition. Today there are over 400 distinct proofs, including a particularly elegant one by James Garfield (later US president) in 1876.

Yes. The 3D distance between two points is d = √((Δx)² + (Δy)² + (Δz)²) — Pythagoras applied twice. The space diagonal of a rectangular box with sides ℓ, w, h is √(ℓ² + w² + h²). More generally, in n-dimensional space, the Euclidean distance is the square root of the sum of squared coordinate differences.