Free Manhattan Distance Calculator

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Manhattan distance (also called taxicab distance or L1 distance) measures distance by summing the absolute differences along each axis, mimicking how a taxi actually has to navigate a city grid — moving only along streets and avenues rather than cutting diagonally through buildings. It gets its name from the rectangular street grid of Manhattan. Beyond navigation, Manhattan distance is widely used in data science and machine learning (as an alternative to Euclidean distance in clustering algorithms like k-means and k-nearest-neighbors), in chess-like grid puzzles (measuring a rook's minimum move count is different, but a king's or pathfinding grid distance often uses this metric), and in error metrics like mean absolute error, which is essentially a Manhattan-distance-based measurement.

How it works

Enter two points (2D or 3D). The calculator finds the absolute difference between the two points' coordinates along each axis separately, then adds those differences together: |x2-x1|+|y2-y1|(+|z2-z1|). Unlike Euclidean distance, there's no squaring or square root involved — it's purely a sum of absolute values.

Enter dimension.
Enter point 1 — x1.
Enter point 1 — y1.
Enter point 1 — z1 (3D only).
Enter point 2 — x2.
Enter point 2 — y2.
Enter point 2 — z2 (3D only).
Click Calculate to see your results.

Examples

(1,2) to (4,6)

|4-1|+|6-2| = 3+4 = 7. (For comparison, the Euclidean straight-line distance here is √(9+16)=5, shorter because it cuts diagonally.)

(0,0,0) to (2,-3,4) in 3D

|2-0|+|-3-0|+|4-0| = 2+3+4 = 9.

Common mistakes to avoid

Forgetting to take absolute values, which could make the "distance" negative or cancel out real separation between the points.
Confusing Manhattan distance with Euclidean distance and expecting the same (shorter) numeric answer.
Applying Manhattan distance to a genuinely diagonal, unconstrained path where Euclidean distance is actually the correct physical measurement.

Frequently asked questions

Euclidean distance is the straight-line "as the crow flies" distance; Manhattan distance follows grid-aligned paths and is always greater than or equal to the Euclidean distance between the same two points.

The Euclidean distance is the length of the direct diagonal path, which by the triangle inequality can never be longer than the sum of the grid-aligned horizontal and vertical legs that make up the Manhattan path.

Whenever movement is genuinely constrained to a grid (city blocks, warehouse robot paths, pixel-grid image processing) or when you specifically want a distance metric less sensitive to large single-axis outliers than squared Euclidean distance.

Yes — only when both points are identical, since every term in the sum is a non-negative absolute value that's zero only when that coordinate matches exactly.

Yes — the same absolute-difference-and-sum pattern extends to any number of dimensions, which is exactly why it's popular in high-dimensional data science applications. This tool specifically supports 2D and 3D.

Yes — whenever the two points differ along only one axis (a purely horizontal or purely vertical move), both distance measures give exactly the same result, since there's no diagonal shortcut available in that case.