What is Pythagorean Theorem?
The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This theorem is widely used in construction, navigation, physics, and computer graphics for calculating distances, determining angles, and solving various spatial problems involving right triangles.
Calculator
² + ² = ²Formula
a² + b² = c²
How to Use
Enter any two known side lengths of a right-angled triangle in the calculator fields. Leave the unknown field empty. Click "Calculate" to find the missing side length. The calculator automatically determines which side to compute based on your inputs. For best results, ensure you enter numerical values only. Use the "Clear" button to reset all fields. Results are displayed with both exact values and decimal approximations for practical applications.
Derivation Process
The Pythagorean theorem was derived by ancient Greek mathematician Pythagoras through geometric analysis. By arranging four identical right triangles within a larger square, he demonstrated that the area of the square formed by the hypotenuse (c²) equals the combined area of squares formed by the other two sides (a² + b²). This geometric proof established the fundamental relationship between the sides of right-angled triangles.
Common Triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
FAQs
1. What is the Pythagorean theorem used for?
The Pythagorean theorem calculates unknown side lengths in right-angled triangles. It's essential in construction, navigation, physics, and computer graphics for determining distances, angles, and spatial relationships.
2. Can Pythagorean theorem be used for non-right triangles?
No, the theorem specifically applies to right-angled triangles. For other triangles, use the Law of Cosines which generalizes the Pythagorean theorem for any triangle.
3. How accurate is the Pythagorean theorem?
The theorem is mathematically exact for Euclidean geometry. Real-world applications' accuracy depends on measurement precision and geometric perfection of the triangle.
4. What are Pythagorean triples?
These are integer combinations satisfying a² + b² = c². Common triples include (3,4,5) and (5,12,13). They represent perfect right-angled triangles with whole number sides.
5. Who invented the Pythagorean theorem?
While named after Pythagoras (570-495 BC), evidence suggests Babylonian mathematicians knew this relationship 1000 years earlier. Pythagoras is credited with the first formal proof.
6. Can the theorem work in 3D space?
Yes, the 3D extension is: d² = a² + b² + c². This calculates space diagonals in rectangular prisms, maintaining the same geometric principle.
7. How is trigonometry related?
Trigonometry extends Pythagorean principles to study triangle relationships. The theorem forms the basis for trigonometric identities and functions like sine and cosine.
8. What's the longest Pythagorean triple?
There are infinitely many triples. Large known primitive triples include (1999, 2000, 2829). Triples grow exponentially larger while maintaining a² + b² = c².
9. Can it prove a triangle is right-angled?
Yes, if sides satisfy a² + b² = c², it must be a right triangle. This converse theorem is used to verify right angles in construction and surveying.
10. How is it used in computer graphics?
The theorem calculates pixel distances, 3D object dimensions, and collision detection. It's fundamental for rendering lighting, shadows, and perspective transformations.