Pythagorean Theorem Calculator
Solve right triangle side lengths using a² + b² = c².
c = √(3² + 4²) = 5 · area 6, perimeter 12
Only right triangles, and only the hypotenuse squared
a² + b² = c² relates the three sides of a right triangle, where c is always the hypotenuse — the side opposite the right angle, and always the longest.Putting a leg where the hypotenuse belongs is the most common mistake: finding a leg means rearranging to b² = c² − a², subtracting rather than adding.
The theorem holds only for right triangles. For any other triangle the equivalent tool is the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras exactly when C is 90° and the cosine term vanishes. The converse is also true and genuinely useful: if a triangle's sides satisfy the equation, the angle opposite c must be a right angle — which is how builders check a corner with a 3-4-5 measurement.
Pythagorean triples are integer solutions: 3-4-5, 5-12-13, 8-15-17, and every multiple of them. The theorem also underpins the distance formula in coordinate geometry, which is just Pythagoras applied to the horizontal and vertical gaps between two points, and it extends to three dimensions asd² = x² + y² + z².
Frequently Asked Questions
What is the Pythagorean theorem formula?
a² + b² = c², where a and b are the perpendicular leg lengths of a right triangle, and c is the hypotenuse.
What are common Pythagorean triples?
Common integer triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.
What is the Pythagorean theorem?
In a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse - the side opposite the right angle and always the longest. Rearranged, a missing leg is sqrt(c^2 - b^2). It applies only to right triangles; the law of cosines generalises it to any triangle.
What are Pythagorean triples?
Sets of three integers satisfying the theorem: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple of a triple is also a triple, so 6-8-10 works too. Builders use the 3-4-5 relationship to square up corners without a set square.