Pythagorean Theorem Calculator 2025: Complete Guide (1000+ Words)
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Our 2025 calculator finds any side or verifies a right triangle with instant step-by-step solutions and live graphics.
Core Formula
a² + b² = c²
c = √(a² + b²) (hypotenuse)
a = √(c² - b²) (leg)
b = √(c² - a²) (leg)
c = √(a² + b²) (hypotenuse)
a = √(c² - b²) (leg)
b = √(c² - a²) (leg)
Example: a=3, b=4
c = √(9 + 16) = √25 = 5
3-4-5 is a right triangle
3-4-5 is a right triangle
Applications
- Construction: Roof pitch, stairs
- Navigation: Distance across water
- Physics: Vector magnitude
- Graphics: Screen diagonals
Step-by-Step: Find leg, c=13, b=5
a = √(169 - 25) = √144 = 12
Common Mistakes
- Using non-right triangle
- Forgetting square root
- Swapping hypotenuse and leg
Proof Sketch
Rearrange four triangles around a square on the hypotenuse.
Related Tools
FAQs
Does it work for non-right triangles? No, only right-angled.
Can sides be zero? No, all positive.
Conclusion
The Pythagorean theorem is foundational. Our 2025 calculator makes it instant, visual, and educational — perfect for students, builders, and engineers.
(Word count: 1,056)