manhattan distance calculator - Aaron Graves, PhDude Replica

Interactive Manhattan Distance Calculator

Enter two points as comma- or space-separated coordinates. This tool supports 2D, 3D, and higher-dimensional vectors.

Enter coordinates for Point A and Point B, then click Calculate Distance.

What Is Manhattan Distance?

Manhattan distance is a way to measure how far apart two points are by moving only along horizontal and vertical paths. Instead of drawing a straight line between points, you travel in grid steps, like walking city blocks in Manhattan. That is why it is also called taxicab distance, city block distance, or L1 distance.

If your movement is constrained to a grid (roads, warehouse aisles, game tiles, matrix coordinates), Manhattan distance often reflects reality better than straight-line distance.

Manhattan Distance Formula

General n-dimensional formula

For two points A and B with coordinates: A = (a₁, a₂, ..., a_n) and B = (b₁, b₂, ..., b_n), the Manhattan distance is:

D(A,B) = |a₁ - b₁| + |a₂ - b₂| + ... + |a_n - b_n|

2D formula

D = |x₁ - x₂| + |y₁ - y₂|

3D formula

D = |x₁ - x₂| + |y₁ - y₂| + |z₁ - z₂|

How to Use This Calculator

Enter the coordinates for Point A in the first field.
Enter the coordinates for Point B in the second field.
Use commas or spaces between values (both work).
Click Calculate Distance to get the result and formula breakdown.
Use Load Example to see a ready-made calculation.

Worked Example

Example (2D)

Suppose A = (1, 2) and B = (5, -1). Manhattan distance = |1 - 5| + |2 - (-1)| = 4 + 3 = 7.

Example (4D)

A = (3, -2, 8, 1), B = (-1, 5, 4, 9) Distance = |3 - (-1)| + |-2 - 5| + |8 - 4| + |1 - 9| = 4 + 7 + 4 + 8 = 23.

When Manhattan Distance Is Useful

Grid navigation: Robots, delivery routing, and game map movement.
Machine learning: KNN and clustering with sparse or high-dimensional data.
Computer vision: Pixel-based feature comparisons where absolute differences matter.
Operations research: Facility layout and warehouse travel optimization.
Urban modeling: Estimating travel across street grids.

Manhattan vs Euclidean Distance

Key difference

Euclidean distance is a straight line (“as the crow flies”), while Manhattan distance is the sum of coordinate-by-coordinate movement. On perfect grids, Manhattan distance can be a better practical measure.

Euclidean (L2): better for direct geometric distance.
Manhattan (L1): better for axis-aligned movement and robust absolute-difference comparisons.

Common Input Mistakes

Using different numbers of dimensions between Point A and Point B.
Entering text that is not numeric (for example: three instead of 3).
Adding extra separators without numbers.
Forgetting negative signs where needed.

FAQ

Can this calculator handle decimals?

Yes. You can enter integers or decimals (for example, 2.5, -1.75, 0).

Can I calculate 5D or 10D Manhattan distance?

Absolutely. As long as both points have the same number of coordinates, this calculator supports any dimension.

Is Manhattan distance always larger than Euclidean distance?

For the same two points, Manhattan distance is always greater than or equal to Euclidean distance. They are equal only in special cases, such as movement along one axis.