Key Takeaways
- The Pythagorean Theorem states that a² + b² = c² for all right triangles
- Side c (hypotenuse) is always the longest side, opposite the 90° angle
- Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
- Used in construction, navigation, GPS technology, and computer graphics
- To find a leg: a = √(c² - b²) or b = √(c² - a²)
What Is the Pythagorean Theorem?
The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right triangle. Named after the ancient Greek mathematician Pythagoras (c. 570-495 BCE), though evidence suggests the relationship was known to Babylonians and Indians centuries earlier.
In any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This elegant relationship has been used for over 2,500 years in construction, navigation, and mathematics.
a² + b² = c²
Classic Example: The 3-4-5 Triangle
Check: 3² + 4² = 9 + 16 = 25 = 5² ✓
How to Use This Calculator
Our Pythagorean Theorem calculator makes finding any side of a right triangle simple:
- Enter any two known sides - Input values for two of the three sides (a, b, or c)
- Leave the unknown side blank - The calculator will solve for the missing value
- Click Calculate - Get instant results with step-by-step solution
- View the visualization - See your triangle drawn to scale
Pro Tip: Always Identify the Hypotenuse First
The hypotenuse (c) is always the longest side and is opposite the right angle. If you're given a word problem, identifying the hypotenuse first helps you set up the equation correctly.
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy the theorem. These are invaluable for quick mental calculations and checking work.
| Triple | Side A | Side B | Hypotenuse | Verification |
|---|---|---|---|---|
| 3-4-5 | 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5-12-13 | 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8-15-17 | 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7-24-25 | 7 | 24 | 25 | 49 + 576 = 625 ✓ |
| 20-21-29 | 20 | 21 | 29 | 400 + 441 = 841 ✓ |
Real-World Applications
The Pythagorean Theorem is used daily in countless practical applications:
- Construction: Ensuring corners are square (the 3-4-5 method)
- Navigation: Calculating distances on maps and GPS systems
- Architecture: Designing roofs, stairs, and structural supports
- Computer Graphics: Calculating distances between pixels
- Surveying: Measuring land and determining property boundaries
- Sports: Calculating diagonal passes and field measurements
Frequently Asked Questions
The Pythagorean Theorem is used to find the length of any side of a right triangle when you know the other two sides. It's essential in construction (ensuring square corners), navigation (calculating distances), architecture (designing roofs and stairs), computer graphics, and many engineering applications.
To find the hypotenuse (c), use the formula: c = √(a² + b²). Square both legs, add them together, then take the square root. For example, if a=3 and b=4: c = √(9 + 16) = √25 = 5.
To find a missing leg, rearrange the formula: a = √(c² - b²) or b = √(c² - a²). Square the hypotenuse, subtract the square of the known leg, then take the square root. For example, if c=13 and a=5: b = √(169 - 25) = √144 = 12.
No, the Pythagorean Theorem only works for right triangles (triangles with one 90-degree angle). For other triangles, you would use the Law of Cosines: c² = a² + b² - 2ab·cos(C), which is a generalization of the Pythagorean Theorem.
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is 3-4-5. Others include 5-12-13, 8-15-17, and 7-24-25. Any multiple of a Pythagorean triple is also a triple (e.g., 6-8-10).
Builders use the 3-4-5 rule to create perfect right angles. Measure 3 feet along one edge and 4 feet along the other. If the diagonal between these points is exactly 5 feet, the corner is square (90 degrees). This method has been used for thousands of years.
Master Right Triangle Calculations
Use our free Pythagorean Theorem calculator for instant, accurate results with step-by-step solutions.