What is a Pythagorean Theorem Calculator?
This calculator finds the third side of a right triangle from the other two sides.
How it's calculated
a² + b² = c² (c is the hypotenuse)
Step-by-step example
A right triangle with legs of 6 and 8 — find the hypotenuse
- Step 1 Apply the formula: c² = a² + b² = 6² + 8²
- Step 2 Calculate: c² = 36 + 64 = 100
- Step 3 c = √100 = 10
Interpretation: This is 2x the classic 3-4-5 triple (6-8-10), a well-known whole-number Pythagorean triple.
The 3-4-5 rule used on construction sites
Measure 3m along one side and 4m along a perpendicular side, and if the diagonal between them is exactly 5m, you have a right angle (since 3²+4²=25=5²). This lets you create a perfect right angle using nothing but a tape measure, no protractor required — a technique used continuously from ancient Egypt to modern construction sites.
Good to know
- Evidence predates Pythagoras by over 1,000 years — the Babylonian clay tablet Plimpton 322 already lists Pythagorean triples, and India and China developed similar results independently.
- Over 400 different proofs exist — including one devised by Einstein at age 12, and another by future US President James Garfield.
- 3-4-5, 5-12-13, and 8-15-17 are examples of Pythagorean triples — sets of three whole numbers satisfying the theorem, and infinitely many exist.
- This theorem is where irrational numbers were first discovered. The diagonal of a unit square is √2, which cannot be expressed as a fraction — a genuine shock to the Pythagoreans, who believed all quantities were ratios of whole numbers.
- Raising the exponent to 3 or higher makes whole-number solutions impossible — this is Fermat's Last Theorem, unsolved for 358 years until finally proven in 1995.
Frequently asked questions
Q. How do I check if a triangle is a right triangle?
Measure all three sides. If a² + b² = c² holds (c being the longest side), it's a right triangle. Construction sites often use the 3:4:5 ratio directly.
Q. How do I find a leg instead of the hypotenuse?
Use a = √(c² − b²). The hypotenuse is always the longest side.
Q. Does this extend to 3D?
Yes. The 3D diagonal is √(a² + b² + c²), used for calculating a box's diagonal length.