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 converse states that if a² + b² = c² holds for a triangle's three sides, the triangle must be right-angled. This is how the theorem is used to test a triangle rather than solve it — exactly the check used in the 'is it right-angled' method above.
It underlies the distance formula between two points on a graph, is used by carpenters and builders to square corners (the classic 3-4-5 method), by surveyors calculating land distances, and in navigation and computer graphics to compute straight-line distances between coordinates.
Yes, extended as d = √(a² + b² + c²) for the space diagonal of a rectangular box — sometimes called the 3D distance formula. It applies the same idea across three perpendicular dimensions instead of two.
Built and maintained by Sanjay Singh · Founder, Numrexo
Builds and maintains the calculators on Numrexo. Each tool's formula is documented on its page, and any reference source it relies on is listed under Method and references.
Last updated: · Calculations run in your browser. Nothing you enter is sent to us or stored.
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