Euclidean Distance Calculator

Euclidean distance is the straight-line distance between two points in space. It is the length of the shortest path connecting them, computed as the square root of the sum of squared coordinate differences: √((x₂−x₁)² + (y₂−y₁)²) in 2D. It generalises to any number of dimensions using the same formula.

For two points A=(a₁, a₂, …, aₙ) and B=(b₁, b₂, …, bₙ): d = √(Σ(aᵢ−bᵢ)²). In 2D: d = √((x₂−x₁)² + (y₂−y₁)²). In 3D: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). This is also called the L2 norm of the difference vector.

Euclidean distance is the straight-line ('as the crow flies') path. Manhattan distance follows axis-aligned grid paths — only horizontal and vertical moves. Euclidean is always ≤ Manhattan. For a 45° diagonal move, Manhattan distance is √2 ≈ 1.41× the Euclidean distance. Use our Manhattan distance calculator to compare both simultaneously.

Use Euclidean when: movement in any direction is equally possible (open space, physics, graphics). Use Manhattan when: movement is grid-constrained (city navigation, certain ML tasks). In machine learning, Manhattan distance can outperform Euclidean in high-dimensional spaces because it's less dominated by large individual differences. For high dimensions, also consider Chebyshev distance.

√(3² + 4²) = √(9+16) = √25 = 5. This is the classic 3-4-5 right triangle. The Manhattan distance between the same points is 3+4 = 7. The ratio is 5/7 ≈ 71.4%.

L2 distance is another name for Euclidean distance. The 'L' refers to Lp norms: L1 is Manhattan (sum of absolute differences), L2 is Euclidean (square root of sum of squares), L∞ is Chebyshev (maximum absolute difference). In machine learning, L2 regularisation (Ridge regression) penalises the L2 norm of model weights.

Euclidean distance is the default metric in k-nearest neighbours (KNN), k-means clustering, and many other algorithms. It measures how similar two data points are — closer points are more similar. It also underlies support vector machines (maximising margin is equivalent to minimising L2 norms) and PCA (which minimises L2 reconstruction error).

For points A=(x₁,y₁,z₁) and B=(x₂,y₂,z₂): d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Example: A=(1,2,3) and B=(4,6,3): d = √(9+16+0) = √25 = 5. In physics, this is simply the distance between two positions in 3D space.

Euclidean distance is a special case of Minkowski distance with p=2. The Minkowski formula is (Σ|aᵢ−bᵢ|ᵖ)^(1/p). At p=1 this becomes Manhattan distance, at p=2 it becomes Euclidean, and as p→∞ it approaches Chebyshev distance. Minkowski generalises all three into a single parameterised family.