What is Manhattan distance?

Manhattan distance (also called taxicab or L1 distance) between two points is the sum of the absolute differences of their coordinates: |x1−x2| + |y1−y2| + … It measures how far apart two points are when movement is restricted to axis-aligned steps, like navigating a grid of city blocks.

How does Manhattan distance differ from Euclidean distance?

Euclidean distance is the straight-line ("as the crow flies") distance computed as √∑(Pi−Qi)². Manhattan distance follows grid paths and is always ≥ Euclidean distance for the same pair of points. Manhattan is less sensitive to large individual-dimension differences, which can make it preferable for high-dimensional or sparse data.

Can I use vectors with more than 2 or 3 dimensions?

Yes. Both formulas generalize to any number of dimensions. Just add more comma-separated values; both inputs must have the same dimension count.

When should I use Manhattan distance over Euclidean?

Manhattan distance (L1) is preferred when dimensions have different units and you want equal weight per axis, when data is high-dimensional and sparse (e.g., text feature vectors), or when you want robustness to outliers in individual features. Euclidean (L2) is more natural for physical 2D/3D space and when direction matters as much as magnitude.

The L1 norm of a vector is the sum of the absolute values of its components: |v1| + |v2| + … Manhattan distance between P and Q equals the L1 norm of the difference vector (P − Q).

Manhattan Distance Calculator

Calculate the Manhattan (taxicab/L1) distance between two points or vectors of any dimension, with Euclidean distance shown for comparison.

Vector Inputs

Point P (comma-separated)

Point Q (comma-separated)

Quick Presets

Manhattan Distance

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L1 / taxicab / city-block

Euclidean Distance

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L2 / straight-line

Per-Dimension Breakdown

Enter values above and click Calculate.

Formula

d(P, Q) = |P₁−Q₁| + |P₂−Q₂| + … + |Pₙ−Qₙ|

Summary

Manhattan distance (also called taxicab or L1 distance) sums the absolute differences of each coordinate pair. It differs from Euclidean distance, which uses the straight-line path; Manhattan distance only allows movement along axis-aligned steps — like navigating city blocks. This tool handles vectors of any dimension and shows both metrics side by side so you can compare them for your use case.

How it works

Enter the coordinates of the first point (P) as a comma-separated list, e.g. "1, 2, 3".
Enter the coordinates of the second point (Q) with the same number of dimensions.
The tool computes |P1−Q1| + |P2−Q2| + … for each dimension to produce the Manhattan distance.
Euclidean distance (√∑(Pi−Qi)²) is computed alongside for comparison.
Component-level breakdown shows the absolute difference for each dimension.

Use cases

Compare feature vectors in machine-learning distance metrics (k-NN, k-means).
Calculate grid-based pathfinding costs in games or robotics.
Check distance between data points when outliers should not be over-penalized (L1 vs L2).
Verify algorithm output during development of spatial or clustering code.
Teach the difference between L1 and L2 norms in linear algebra coursework.
Compute pixel-wise L1 distance between image patches in computer vision.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu