Solve for any side of a right triangle using a² + b² = c². Check if three lengths form a right triangle, identify Pythagorean triples, generate triples with Euclid's formula, and explore the classic geometric proof.
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Lists all primitive and non-primitive triples containing this leg (search up to 10×value).
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The classic visual proof: arrange four congruent right triangles inside a square of side c. The inner square has side (b − a), so c² = 4·(½ab) + (b − a)² = 2ab + b² − 2ab + a² = a² + b².
Diagram updates when you calculate a triangle.
Theorem, Converse & History
a² + b² = c²
Theorem: In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Converse: If a² + b² = c² for a triangle's sides, then the angle opposite c is exactly 90°.
History: Despite the name, Pythagorean triples were known to Babylonian mathematicians over 1,000 years before Pythagoras (~570–495 BC). The Plimpton 322 clay tablet (~1800 BC) lists 15 Pythagorean triples. Pythagoras (or his school) is credited with the first general proof.
Connection to Trigonometry
sin²θ + cos²θ = 1
The Pythagorean identity sin²θ + cos²θ = 1 is simply the Pythagorean theorem applied to a unit circle.
On a unit circle, a point at angle θ has coordinates (cos θ, sin θ). These are the legs of a right triangle with hypotenuse 1, so: cos²θ + sin²θ = 1².
Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives cot²θ + 1 = csc²θ — the other two Pythagorean identities.
Our tutors break down right triangles, trig ratios, and identities step by step — at your pace.