Euclidean Distance Calculator
Calculate the straight-line distance between points in 2D or 3D space
2D Coordinate Points
Enter coordinates for two points in 2D space
2D Results
Distance and geometric properties
2D Examples
Load common 2D coordinate examples
Understanding Euclidean Distance
Euclidean distance is the most common way to measure the straight-line distance between two points in space. Named after the ancient Greek mathematician Euclid, this distance metric represents the shortest possible path between two points and forms the foundation of classical geometry and many modern applications in mathematics, physics, and computer science.
Distance Formulas
2D Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
For points (x₁, y₁) and (x₂, y₂) in a 2D plane
3D Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
For points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space
Mathematical Foundation
The Euclidean distance formula is derived from the Pythagorean theorem. In 2D space, we create a right triangle where the distance between two points forms the hypotenuse:
- Calculate the horizontal distance: Δx = x₂ - x₁
- Calculate the vertical distance: Δy = y₂ - y₁
- Apply the Pythagorean theorem: d² = Δx² + Δy²
- Take the square root to find the distance: d = √(Δx² + Δy²)
For 3D space, we extend this concept by adding the third dimension (z-axis), creating a three-dimensional right triangle where the distance is the space diagonal.
Related Geometric Properties
Midpoint
The point exactly halfway between two points
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope (2D)
The rate of change between two points
m = (y₂-y₁)/(x₂-x₁)
Direction Vector (3D)
Vector pointing from one point to another
v⃗ = ⟨Δx, Δy, Δz⟩
Real-World Applications
Navigation & GPS
Calculating shortest paths, distance estimation, and route optimization.
Computer Graphics
3D modeling, collision detection, and spatial transformations.
Machine Learning
K-nearest neighbors, clustering algorithms, and similarity measures.
Physics & Engineering
Force calculations, motion analysis, and structural design.
Distance Metrics Comparison
| Metric | Formula (2D) | Use Case |
|---|---|---|
| Euclidean | √[(x₂-x₁)² + (y₂-y₁)²] | Straight-line distance, physical measurements |
| Manhattan | |x₂-x₁| + |y₂-y₁| | Grid-based movement, city blocks |
| Chebyshev | max(|x₂-x₁|, |y₂-y₁|) | Chess moves, maximum coordinate difference |
Worked Examples
2D Example: Points (0, 0) and (3, 4)
Point 1: (0, 0)
Point 2: (3, 4)
Δx = 3 - 0 = 3
Δy = 4 - 0 = 4
Distance = √(3² + 4²) = √(9 + 16) = √25 = 5
This is the famous 3-4-5 right triangle!
3D Example: Points (0, 0, 0) and (1, 1, 1)
Point 1: (0, 0, 0)
Point 2: (1, 1, 1)
Δx = 1, Δy = 1, Δz = 1
Distance = √(1² + 1² + 1²) = √3 ≈ 1.732
Space diagonal of a unit cube
Advanced Concepts
Higher Dimensions
The Euclidean distance formula extends to any number of dimensions: d = √(Σ(xᵢ₂ - xᵢ₁)²) where the sum is over all dimensions.
Weighted Euclidean Distance
Different dimensions can be given different importance using weights: d = √(Σwᵢ(xᵢ₂ - xᵢ₁)²) where wᵢ are the weights.
Computational Considerations
For performance-critical applications, squared Euclidean distance (without the square root) is often used since it preserves ordering relationships.