Pythagorean Theorem
The definitive tool for right triangle calculations.
Pythagorean Theorem Calculator
The Pythagorean theorem is one of the most famous equations in mathematics: a² + b² = c². Enter any two sides of a right triangle and this calculator finds the third instantly.
How to Use This Calculator
- Enter any two sides of the right triangle (legs a and b, or hypotenuse c).
- Leave the unknown side blank.
- Click Calculate to find the missing side, angle, area, and perimeter.
The Pythagorean Theorem
a² + b² = c²
- a, b = legs (the two shorter sides forming the right angle)
- c = hypotenuse (the longest side, opposite the right angle)
Finding hypotenuse: c = √(a² + b²)
Finding a leg: a = √(c² − b²)
Example Calculations
Classic 3-4-5 right triangle: a=3, b=4 → c = √(9+16) = √25 = 5
a=5, b=12 → c = √(25+144) = √169 = 13
c=10, a=6 → b = √(100−36) = √64 = 8
Pythagorean Triples
Integer combinations that satisfy a²+b²=c²:
- 3, 4, 5 (and multiples: 6,8,10 — 9,12,15 — etc.)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- 20, 21, 29
Real-World Applications
- Construction: Checking if corners are square (3-4-5 method)
- Navigation: Finding straight-line distance from two perpendicular directions
- Screen size: A 16:9 screen with 27-inch diagonal → √(16²+9²) = √337 ≈ 18.36 units per unit → each unit = 27/18.36 ≈ 1.47 inches
- Rope ladders and staircases: Finding ladder length from height and distance
Common Mistakes to Avoid
- Applying it to non-right triangles — The theorem only works for triangles with exactly one 90° angle. For other triangles, use the law of cosines.
- Making the longest side a or b instead of c — c must be the hypotenuse (side opposite the right angle). The equation fails if assigned wrong.
- Forgetting to take the square root — c² = a² + b² gives you c squared. Take the square root to get c.
Frequently Asked Questions
Who proved the Pythagorean theorem?
Though named after Pythagoras (~570–495 BC), the theorem was known to Babylonian and Indian mathematicians centuries earlier. Pythagoras (or his followers) is credited with the first formal proof in Western mathematics.
Does this work in 3D?
For a box with dimensions l, w, h, the space diagonal = √(l²+w²+h²). This is an extension of the theorem applied twice.
What is a converse of the Pythagorean theorem?
If a²+b²=c², then the triangle is a right triangle. This lets you verify right angles in construction and carpentry.
Conclusion
The Pythagorean theorem is one of the oldest and most useful mathematical relationships. Use this calculator for any right triangle problem — construction, navigation, geometry homework, or everyday measurement.
Related: Right Triangle Calculator | Triangle Calculator | Distance Calculator
Expert Tip
Use the 3-4-5 rule in construction to check if a corner is square. Measure 3ft on one wall and 4ft on the other; the distance between the marks must be exactly 5ft.
