Pythagorean Theorem Calculator
Calculate hypotenuse c using the formula: c = √(a² + b²)
The Pythagorean Theorem Explained
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental principle in geometry that states: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Expressed as an equation: a² + b² = c²
Where:
- a and b are the lengths of the legs (the two sides that form the right angle)
- c is the length of the hypotenuse (the side opposite the right angle)
How to Use This Calculator
- Select which side you want to calculate (a, b, or c).
- Enter the values for the other two sides.
- Choose the unit of measurement.
- Click "Calculate" to see the result.
Derivations from the Pythagorean Theorem
To find a: a = √(c² - b²)
To find b: b = √(c² - a²)
To find c: c = √(a² + b²)
Applications of the Pythagorean Theorem
Practical Uses
- Construction and Carpentry: Ensuring corners are square, calculating diagonal measurements.
- Navigation: Calculating distances when traveling in perpendicular directions.
- Engineering: Designing structures, calculating forces in perpendicular components.
- Computer Graphics: Calculating distances between points in 2D and 3D space.
- Surveying: Measuring land and determining boundaries.
Real-World Examples
Example 1: A ladder (c) leaning against a wall forms a right triangle with the ground (b) and the wall (a). If the ladder is 5 meters long and the base is 3 meters from the wall, how high up the wall does the ladder reach?
Using a² + b² = c², we get: a² + 3² = 5², so a² = 25 - 9 = 16, and a = 4 meters.
Example 2: You're planning to install a diagonal brace in a rectangular frame that is 6 feet wide and 8 feet tall. How long should the brace be?
Using c² = a² + b², we get: c² = 6² + 8² = 36 + 64 = 100, so c = 10 feet.
FAQ
Does the Pythagorean theorem work for all triangles?
No, the Pythagorean theorem only applies to right triangles (triangles with one 90° angle).
What are Pythagorean triples?
Pythagorean triples are sets of three integers that satisfy the Pythagorean theorem. The most famous example is (3, 4, 5) since 3² + 4² = 9 + 16 = 25 = 5². Other examples include (5, 12, 13), (8, 15, 17), and (7, 24, 25).
Who discovered the Pythagorean theorem?
The theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BCE), although there is evidence that the relationship was known to earlier civilizations, including the Babylonians and Chinese.
How accurate are the calculations?
This calculator performs computations with high precision but rounds the displayed results to four decimal places for readability.
Common Pythagorean Triples
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 3² + 4² = 9 + 16 = 25 = 5² |
| 5 | 12 | 13 | 5² + 12² = 25 + 144 = 169 = 13² |
| 8 | 15 | 17 | 8² + 15² = 64 + 225 = 289 = 17² |
| 7 | 24 | 25 | 7² + 24² = 49 + 576 = 625 = 25² |
| 9 | 40 | 41 | 9² + 40² = 81 + 1600 = 1681 = 41² |