The Pythagorean theorem states that in a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. It is one of the most famous results in mathematics, with over 400 known proofs.
The Formula
Given legs a and b and hypotenuse c: c = sqrt(a^2 + b^2). Try our Pythagorean Theorem Calculator for instant results.
Visual Proof Sketch
Arrange four copies of a right triangle around a square. The area of the large square equals the sum of areas of two smaller squares built on the legs - demonstrating a^2 + b^2 = c^2.
Applications
Construction workers use it to ensure corners are square. Navigation systems compute distances. Computer graphics rely on it for vector magnitudes.
Pythagorean Triples
Integer solutions include (3,4,5), (5,12,13), and (8,15,17). Recognizing triples speeds up mental math.
Going Deeper
Explore the Pythagorean theorem with proofs, examples, and real-world applications. This guide connects theory to practice — use the related calculators linked at the bottom to verify each example with your own numbers.
Practical Tips
- Write down given values and unknowns before opening the calculator.
- Check units and rounding rules appropriate to your context (class, lab, or business).
- Compare manual working with the calculator result to build confidence.
Common Mistakes to Avoid
- Rushing inputs without reading field labels carefully.
- Confusing similar formulas that use different variables or units.
- Reporting results with more precision than your inputs justify.
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