What is the Manhattan distance between two points?

The Manhattan distance is the sum of the absolute differences of the coordinates: d₁ = Σ |aᵢ − bᵢ|. It is the distance you would travel between two points along a grid of streets, never diagonally — which is why it is also called the taxicab or city-block distance. For the points (2, 3) and (5, 7) it is |2 − 5| + |3 − 7| = 3 + 4 = 7.

How do you calculate L1 distance?

Pair the two vectors coordinate by coordinate, take the absolute value of each difference, and add them up. In symbols, for vectors a and b of the same length: L1 = |a₁ − b₁| + |a₂ − b₂| + … + |aₙ − bₙ|. There is no squaring and no square root, so it is faster to compute than Euclidean distance and less sensitive to a single large coordinate gap.

What is the difference between Manhattan and Euclidean distance?

Manhattan distance, Σ |aᵢ − bᵢ|, measures the path along axis-aligned moves, like walking a city grid. Euclidean distance, √(Σ (aᵢ − bᵢ)²), measures the straight-line, as-the-crow-flies length. Manhattan is always greater than or equal to Euclidean. For (2, 3) to (5, 7), Manhattan is 7 while Euclidean is 5 — the grid detour costs two extra units.

When should I use Manhattan distance instead of Euclidean?

Use Manhattan when movement is restricted to axes (grid maps, chip routing), when features are on different scales and you want to limit the influence of any single large difference, or in high-dimensional data where L1 often separates points more reliably than L2. In KNN and k-means, switching metric='manhattan' can change which neighbours are nearest, so it is worth comparing both.

Is Manhattan distance the same as taxicab geometry?

Yes. Taxicab geometry is the metric space where distance is measured by Manhattan (L1) distance instead of Euclidean. The name comes from a taxi driving on a rectangular street grid: it cannot cut diagonally, so the trip length is the sum of the horizontal and vertical blocks travelled. City-block distance and L1 distance are the same quantity under different names.

Is Manhattan distance the same as Hamming distance?

For binary vectors (each coordinate 0 or 1) they are equal: each differing position contributes exactly 1 to both. For a = [1, 0, 1, 1] and b = [0, 0, 1, 0] both give 2. For general numeric vectors they differ — Manhattan adds the magnitudes of the differences, while Hamming only counts how many positions differ regardless of size.

Does this match scikit-learn manhattan_distances and SciPy cityblock?

Yes. The L1 output here equals sklearn.metrics.pairwise.manhattan_distances and scipy.spatial.distance.cityblock, and the comparison row matches euclidean_distances and the DistanceMetric "chebyshev". The tool also recomputes the distance a second way, via the general Minkowski p = 1 formula (how scipy.spatial.distance.minkowski works), and confirms the two routes agree.

Does this calculator send my data anywhere?

No. Parsing your numbers, taking the differences, the absolute values and the sum all run in your browser with plain JavaScript. Nothing is uploaded, logged, or stored, so you can paste real feature vectors or embedding values without concern. The page keeps working offline once it has loaded.

AI · Machine learning

Manhattan Distance Calculator

Compute the Manhattan (L1 / taxicab / city-block) distance between two points or two vectors of any dimension, in your browser. See the full per-dimension working, the substituted expression, and the Euclidean and Chebyshev comparisons. Matches scikit-learn and SciPy — no signup, nothing uploaded.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jun 11, 2026

Manhattan distance calculator

Input mode

Vector / Point A

Any length ≥ 1 — A and B must match.

Vector / Point B

Same length as A. Commas, spaces, or new lines.

Examples

Manhattan (L1)

5.0000

Σ |aᵢ − bᵢ| · 3-D

Euclidean (L2)

3.6056

√(Σ (aᵢ − bᵢ)²)

Chebyshev (L∞)

3.0000

max |aᵢ − bᵢ|

Decimals

Worked expression

|1.0000 − 4.0000| + |2.0000 − 0.0000| + |3.0000 − 3.0000| = 3.0000 + 2.0000 + 0.0000 = 5.0000

Cross-check. The direct form Σ |aᵢ − bᵢ| gives 5.0000; the independent general Minkowski p = 1 form (Σ |aᵢ − bᵢ|¹)^(1/1) — how SciPy's minkowski(u, v, 1) computes it — gives 5.0000. They reconcile, as they must.

Step-by-step working

Dim	aᵢ	bᵢ	aᵢ − bᵢ	\|aᵢ − bᵢ\|	(aᵢ − bᵢ)²
#1	1.0000	4.0000	-3.0000	3.0000	9.0000
#2	2.0000	0.0000	2.0000	2.0000	4.0000
#3	3.0000	3.0000	0.0000	0.0000	0.0000
Totals				5.0000	13.0000

Method: Manhattan d₁ = Σ |aᵢ − bᵢ|; Euclidean d₂ = √(Σ (aᵢ − bᵢ)²); Chebyshev d∞ = maxᵢ |aᵢ − bᵢ| — scikit-learn manhattan_distances and SciPy cityblock. Nothing leaves this page.

How it works

Manhattan distance — also called L1, taxicab, or city-block distance — is the distance between two points measured along axis-aligned moves, the way a taxi travels a rectangular street grid rather than cutting diagonally. For two equal-length vectors a = [a₁…aₙ] and b = [b₁…bₙ], it is the sum of the absolute coordinate differences. It is the metric scikit-learn calls manhattan_distances and SciPy calls cityblock.

d₁(a, b) = Σ |aᵢ − bᵢ| = |a₁ − b₁| + … + |aₙ − bₙ|

The tool computes this in three steps:

Per-dimension difference. Subtract the vectors coordinate by coordinate: dᵢ = aᵢ − bᵢ.
Absolute value and sum. Take |dᵢ| for each dimension and add them. That sum Σ |dᵢ| is the Manhattan distance — no squaring, no square root.
Comparison metrics. Over the same differences the tool also reports Euclidean √(Σ dᵢ²) and Chebyshev max |dᵢ|, the L2 and L∞ members of the same Minkowski family.

All three metrics are special cases of the Minkowski distance dₚ = (Σ |aᵢ − bᵢ|ᵖ)^(1/p): p = 1 gives Manhattan, p = 2 gives Euclidean, and p → ∞ gives Chebyshev. Because Manhattan is the p = 1 case, the calculator uses that general form as an independent cross-check — it recomputes (Σ |dᵢ|¹)^(1/1) and confirms it reconciles with the direct Σ |dᵢ| sum, exactly as SciPy's minkowski(u, v, 1) equals cityblock(u, v). All arithmetic is double-precision and is rounded only for display, using the decimal places you pick, so even inputs around 10⁹ stay exact.

Worked examples

n-D vectors — a = [1, 2, 3], b = [4, 0, 3]

Differences: 1 − 4 = −3, 2 − 0 = 2, 3 − 3 = 0
Absolute values: 3, 2, 0
Manhattan: |−3| + |2| + |0| = 3 + 2 + 0 = 5
Euclidean: √(9 + 4 + 0) = √13 = 3.605551, Chebyshev: max(3, 2, 0) = 3
cityblock([1,2,3], [4,0,3]) = 5.0 → matches

2D points, L1 vs L2 — a = [2, 3], b = [5, 7]

Differences: 2 − 5 = −3, 3 − 7 = −4
Absolute values: 3, 4
Manhattan: |−3| + |−4| = 3 + 4 = 7
Euclidean: √(9 + 16) = √25 = 5, Chebyshev: max(3, 4) = 4
The grid path (7) is longer than the straight line (5)

Binary vectors, Manhattan = Hamming — a = [1, 0, 1, 1], b = [0, 0, 1, 0]

Differences: 1, 0, 0, 1
Absolute values: 1, 0, 0, 1
Manhattan: 1 + 0 + 0 + 1 = 2
Euclidean: √2 = 1.414214, Chebyshev: 1
Equals the Hamming distance (2 differing positions) for 0/1 data

Frequently asked questions

Sources & references

The formulas on this page were last cross-checked against these sources on 2026-06-11. Manhattan distance is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.

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