Formula & Calculator
Pythagorean Theorem
Relates the lengths of the two legs of a right triangle to its hypotenuse, one of the oldest and most widely used theorems in mathematics.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| c | Length of the hypotenuse | |
| a, b | Lengths of the two legs of the right triangle |
What it means
The Pythagorean theorem is a fundamental relation in Euclidean geometry: the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is used to compute distances in 2D and 3D (Euclidean distance), to find the magnitude of vectors, and in many practical applications such as construction, navigation, and surveying. The theorem is also the basis for the definition of trigonometric functions. There are hundreds of proofs; it is one of the most important and well‑known theorems in mathematics.
Worked example
Pythagorean Theorem – Two Examples
Real‑World| Parameter | Value |
|---|---|
| a | 3 m |
| b | 4 m |
| Parameter | Value |
|---|---|
| a | 6 ft |
| b | 8 ft |
Common mistakes
- Pythagorean theorem: c = √(a² + b²) – for a right‑angled triangle (c is the hypotenuse).
- Units: a, b, and c must be in the same units.
- Check: Ensure the triangle is right‑angled – if not, use the Law of Cosines.
- 3D: For 3D distances, use the generalised distance formula.
- Common mistake: Forgetting the square root – c² = a² + b² is the squared length.
Applications
The Pythagorean theorem, c² = a² + b² (for a right triangle), is one of the oldest and most widely used mathematical theorems. It is used to compute distances, to check right angles, and to derive many trigonometric identities. Engineers use it in construction for squaring foundations, in surveying for distance measurement, and in navigation for path planning. In physics, it combines perpendicular vector components (e.g., velocity, force). In computer graphics, it calculates distances between points. This theorem is a pillar of geometry and is applied daily in countless practical and theoretical contexts, from simple woodworking to complex structural analysis.
- Distance measurement in surveying, construction, and navigation
- Vector magnitude computation from perpendicular components
- Layout and alignment of buildings and structures
- Derivation of trigonometric identities and angle relationships
- Computer graphics – distance and collision detection