Euclidean Distance Calculator
Calculate the shortest distance between points, lines, and geometric objects in n-dimensional space
Calculate Euclidean Distance
Point Coordinates
Distance Results
Formula: d = √[Σᵢ₌₁ⁿ(qᵢ - pᵢ)²]
Calculation: d = √[(0 - 0)² + (0 - 0)²]
Step-by-step: d = √[0.000 + 0.000] = √0.000 = 0.000000
Example Calculations
2D Distance Example
Points: p = (1, 2) and q = (4, 6)
Calculation: d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5
Result: Distance = 5 units
3D Distance Example
Points: p = (1, 2, 3) and q = (4, 6, 8)
Calculation: d = √[(4-1)² + (6-2)² + (8-3)²] = √[9 + 16 + 25] = √50 ≈ 7.071
Result: Distance ≈ 7.071 units
Point to Line Example
Point: (2, 3)
Line: 3x + 4y - 12 = 0
Calculation: d = |3×2 + 4×3 - 12| / √(3² + 4²) = |6 + 12 - 12| / 5 = 6/5 = 1.2
Result: Distance = 1.2 units
Distance Types
Two Points
Straight-line distance between two points
Works in 1D to 4D space
Three Points
All pairwise distances in a triangle
Forms triangle perimeter
Point to Line
Shortest distance from point to line
Perpendicular distance
Parallel Lines
Distance between parallel lines
Constant separation
Mathematical Tips
Euclidean distance is the shortest path in flat space
Works in any number of dimensions
Also called L2 norm or straight-line distance
Essential in machine learning and data analysis
Understanding Euclidean Distance
What is Euclidean Distance?
The Euclidean distance is the "ordinary" straight-line distance between two points in Euclidean space. It's the length of the line segment connecting two points and represents the shortest path between them in flat (non-curved) space.
Applications
- •Machine learning and clustering algorithms
- •Computer graphics and game development
- •Navigation and GPS systems
- •Statistical analysis and data science
Mathematical Formulation
d(p,q) = √[Σᵢ₌₁ⁿ(qᵢ - pᵢ)²]
General n-dimensional formula
Common Cases:
- 2D: d = √[(x₂-x₁)² + (y₂-y₁)²]
- 3D: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
- Point to Line: d = |ax₀ + by₀ + c| / √(a² + b²)
- Parallel Lines: d = |c₂ - c₁| / √(a² + b²)
Note: The Euclidean distance is always non-negative and equals zero only when comparing identical points.