Pythagorean Theorem
Calculate triangle sides
Pythagorean Theorem: a² + b² = c²
Solve right triangles instantly
Result
About Pythagorean Theorem Calculator
The Pythagorean Theorem Calculator helps you solve right triangles instantly. Whether you need to find the hypotenuse, a missing leg, or check if a triangle is right-angled, our calculator provides accurate results with step-by-step formulas.
Perfect for students, teachers, carpenters, architects, and anyone working with triangles. The Pythagorean theorem (a² + b² = c²) is one of the most fundamental concepts in geometry.
How to Use This Pythagorean Theorem Calculator
Step 1: Select what you want to find: Hypotenuse, Missing Leg, or Check Right Triangle.
Step 2: Enter the known side lengths (legs for hypotenuse, hypotenuse + one leg for missing leg, all three sides for check).
Step 3: Click "Calculate" to see your result.
Step 4: View the calculated side length, formula used, and check if it's a Pythagorean triple.
Step 5: Use the Reset button to clear all inputs and try a different calculation.
Why Use a Pythagorean Theorem Calculator?
✓ Quick Solutions
Get instant answers to right triangle problems. No manual square root calculations needed.
✓ Construction & Carpentry
Check right angles and calculate diagonal lengths. Essential for building, framing, and layout.
✓ Education & Homework
Check your Pythagorean theorem homework answers. Understand the formula with step-by-step solutions.
✓ Multiple Modes
Find hypotenuse, missing leg, or verify right triangles. All in one calculator.
Real-World Applications of Pythagorean Theorem
- 🔨Construction: Checking right angles in buildings, ensuring walls are perpendicular, calculating diagonal bracing lengths.
- 🔨Navigation: Calculating shortest distances between two points (as the crow flies).
- 🔨Sports: Calculating distances in baseball (base paths), football (field goal angles), and golf (shots).
- 🔨Computer Graphics: Calculating distances between pixels, object collision detection, 3D rendering.
Formula
a² + b² = c²
Where c is the hypotenuse (longest side), a and b are the legs
To find hypotenuse: c = √(a² + b²) | To find leg: a = √(c² - b²)
Pythagorean Triples (Integer Solutions)
| a | b | c (hypotenuse) |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
| 9 | 12 | 15 |
| 9 | 40 | 41 |
Frequently Asked Questions
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. Formula: a² + b² = c², where c is the hypotenuse (longest side). Named after Greek mathematician Pythagoras (570-495 BCE).
If you know both legs (a and b), use: c = √(a² + b²). Example: a=3, b=4 → c = √(9+16) = √25 = 5. This is the classic 3-4-5 right triangle, often used in construction for checking right angles.
If you know the hypotenuse and one leg: a = √(c² - b²). Example: c=5, b=4 → a = √(25-16) = √9 = 3. This formula works for any right triangle where you know two sides.
A Pythagorean triple is a set of three integers that satisfy a² + b² = c². Common triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41. These are useful in geometry problems and construction.
Check if a² + b² = c² (where c is the longest side). If true, it's a right triangle. Example: sides 6,8,10 → 36+64=100 → 100=100 → Yes, it's a right triangle.
The Pythagorean Theorem: In any right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. Formula: a² + b² = c². One of the most important theorems in geometry, used in construction, navigation, and physics.
If you know both legs: Hypotenuse = √(a² + b²). Example: a=5, b=12 → c = √(25+144) = √169 = 13. The 5-12-13 triangle is a Pythagorean triple. Our calculator does this instantly.
If you know hypotenuse and one leg: Missing leg = √(c² - a²). Example: c=13, a=5 → b = √(169-25) = √144 = 12. Works for any right triangle with two known sides.
Pythagorean triples are integer solutions to a² + b² = c². Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29. Multiples also work: 6-8-10, 9-12-15. Our calculator identifies if your triangle is a Pythagorean triple.
Check if a² + b² = c² (c = longest side). Example: 8-15-17 → 64+225=289 → 289=289 → Right triangle! If not equal, it's not a right triangle. Use our 'Check Right Triangle' mode for instant verification.