Manhattan Distance Calculator

The Manhattan distance is a way of measuring the distance between two points by summing the absolute differences of their coordinates along each axis. It gets its name from the grid-like street layout of Manhattan, New York — where you can only travel along horizontal and vertical streets, never diagonally. It is also called the taxicab distance, city block distance, or L₁-norm.

For two points A(x₁, y₁) and B(x₂, y₂) in 2D, the Manhattan distance is d = |x₂ − x₁| + |y₂ − y₁|. In 3D it extends to d = |x₂ − x₁| + |y₂ − y₁| + |z₂ − z₁|. More generally, for N dimensions it is the sum of absolute differences across all coordinate pairs.

Euclidean distance is the straight-line ('as the crow flies') distance between two points, calculated using the Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²). Manhattan distance instead travels only along grid-aligned paths, summing absolute coordinate differences. Manhattan distance is always greater than or equal to the Euclidean distance between the same two points.

Manhattan distance is always ≥ Euclidean distance for any two points. The Euclidean distance represents the minimum possible straight-line path, while the Manhattan distance represents travel constrained to a grid. The two are equal only when the two points differ along exactly one coordinate axis (i.e., movement is purely horizontal or purely vertical).

Manhattan distance is widely used in machine learning (e.g., k-nearest neighbors, k-means clustering), data science, robotics path planning, image processing, and operations research. It is preferred over Euclidean distance when movement is grid-constrained, or when robustness to outliers is needed — since it does not square the differences, large deviations have less outsized influence.

The Chebyshev distance (L∞-norm) is the maximum of the absolute differences along any single coordinate axis: d = max(|x₂−x₁|, |y₂−y₁|). It represents the minimum number of moves a king needs on a chessboard. While Manhattan distance sums all axis differences, Chebyshev distance takes only the largest one.

Yes. The Manhattan distance generalises naturally to any number of dimensions by summing the absolute differences of all coordinate pairs. This calculator supports 2D and 3D calculations. In machine learning applications, it is routinely applied to high-dimensional feature vectors.

Yes — the L₁-norm of the difference vector between two points is identical to their Manhattan distance. When you subtract the coordinate vectors of two points and take the L₁-norm (sum of absolute values of all components), you get the same result as the Manhattan distance formula.