Manhattan Distance Calculator – Guide & Formulas
Calculate the Manhattan distance (taxicab distance) between two points. Free online tool for grid-based distance calculations with step-by-step solutions.
Calculate the Manhattan distance between two points on a grid. Enter coordinates to get the taxicab distance with step-by-step explanations and visual comparisons to Euclidean distance.
Key Takeaway
Use the free Manhattan Distance Calculator to calculate the manhattan distance (taxicab distance) between two points. free online tool for grid-based distance calculations with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Manhattan Distance Calculator
- Enter the coordinates of the first point (x₁, y₁).
- Enter the coordinates of the second point (x₂, y₂).
- Click Calculate to compute the Manhattan distance.
- Review the step-by-step calculation comparing it to Euclidean distance.
The Formula
Variable Definitions
- x₁, y₁: Coordinates of the first point
- x₂, y₂: Coordinates of the second point
- d: Manhattan (taxicab) distance between the two points
Distance from (1, 3) to (7, 11)
Calculate the Manhattan distance between two grid points.
- Step 1: Calculate horizontal distance: |7 - 1| = 6.
- Step 2: Calculate vertical distance: |11 - 3| = 8.
- Step 3: Manhattan distance = 6 + 8 = 14 units.
- Step 4: Compare to Euclidean: √(6² + 8²) = √(36+64) = √100 = 10 units.
Frequently Asked Questions
Why is it called Manhattan distance?
The name comes from the grid-like street layout of Manhattan, New York City. To travel between two points in Manhattan, you must follow the grid streets — you cannot cut diagonally through buildings.
How does it differ from Euclidean distance?
Euclidean distance is the straight-line (as-the-crow-flies) distance: √((Δx)² + (Δy)²). Manhattan distance is the distance along grid lines: |Δx| + |Δy|. Manhattan distance is always ≥ Euclidean distance.
What is the L1 norm?
The L1 norm is the sum of absolute values of vector components. Manhattan distance is the L1 norm of the difference vector between two points. Euclidean distance is the L2 norm.
Can I use this in 3D?
Yes. In 3D: d = |x₁-x₂| + |y₁-y₂| + |z₁-z₂|. The principle extends to any number of dimensions.
What is Chebyshev distance?
Chebyshev distance (L∞ norm) is the maximum of the absolute differences: max(|Δx|, |Δy|). It represents the minimum number of king moves on a chessboard.
Where is Manhattan distance used in practice?
It is used in computer science (hash tables, image processing), robotics (grid-based pathfinding), machine learning (L1 regularization), and urban planning (accessibility metrics).
Is Manhattan distance always larger than Euclidean?
Yes, for any two points, the Manhattan distance is always greater than or equal to the Euclidean distance. They are equal only when the points share the same x or y coordinate.
What about Minkowski distance?
Minkowski distance is a generalization: d = (Σ|xᵢ - yᵢ|^p)^(1/p). When p=1, it gives Manhattan; p=2 gives Euclidean; p=∞ gives Chebyshev.