Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean Theorem: a² + b² = c²
Result
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Formula: a² + b² = c²
Where:
- a and b are the lengths of the two legs (the sides that form the right angle)
- c is the length of the hypotenuse (the side opposite the right angle)
Finding the Missing Side
To find a missing side, rearrange the formula:
- To find the hypotenuse (c): c = √(a² + b²)
- To find a leg (a): a = √(c² - b²)
- To find a leg (b): b = √(c² - a²)
Image: Pythagorean Theorem visualization
Example
If a = 3 units and b = 4 units, find the hypotenuse c:
c² = a² + b²
c² = 3² + 4²
c² = 9 + 16
c² = 25
c = √25 = 5 units
Applications of the Pythagorean Theorem
Real-world Applications
- Construction: Ensuring corners are square (3-4-5 rule)
- Navigation: Calculating distances on maps
- Architecture: Designing structures and calculating dimensions
- Physics: Calculating resultant forces and vectors
- Surveying: Measuring distances and heights
Mathematical Applications
- Distance Formula: Finding distance between two points in a coordinate plane
- Trigonometry: Foundation for trigonometric identities
- Geometry: Solving problems involving right triangles
- Vector Calculations: Finding magnitudes of vectors
About Pythagorean Theorem
The Pythagorean Theorem is named after the ancient Greek mathematician Pythagoras (c. 570–495 BC). It is one of the fundamental theorems in geometry and has numerous applications in mathematics, science, and everyday life.
While the theorem is attributed to Pythagoras, there is evidence that the Babylonians and Egyptians were aware of the relationship between the sides of a right triangle much earlier.
The theorem has been proven in many different ways throughout history, with over 350 different proofs documented.
Pythagorean Triples
Pythagorean triples are sets of three positive integers a, b, and c that satisfy the equation a² + b² = c².
The most well-known Pythagorean triple is (3, 4, 5), but there are infinitely many such triples.
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 |