Manhattan Distance Calculator
Calculate the taxicab distance between two points in N-dimensional space
Enter Coordinates
First Point (A)
Second Point (B)
Results
Points
Step-by-Step Calculation
Example: Walking in Manhattan
Scenario
You live at the corner of 2nd Avenue and 9th Street: A(2, 9)
The grocery store is at 3rd Avenue and 5th Street: B(3, 5)
Calculation
Manhattan Distance: d = |2 - 3| + |9 - 5| = 1 + 4 = 5 blocks
Euclidean Distance: d = √[(2-3)² + (9-5)²] = √[1 + 16] = √17 ≈ 4.12 blocks
Result: You need to walk 5 city blocks, not 4.12 blocks in a straight line
Distance Comparison
Manhattan
Sum of absolute differences
|a₁-b₁| + |a₂-b₂| + ...
Euclidean
Straight line distance
√[(a₁-b₁)² + (a₂-b₂)² + ...]
Note: Manhattan distance is always ≥ Euclidean distance
Applications
Urban Planning
City block navigation, taxi routing
Chess
Rook movement distance calculation
Biology
Gene splicing, molecular distances
Gaming
Snake game pathfinding
Machine Learning
Feature similarity, clustering
Tips
Also called taxicab, city block, or snake distance
Useful when movement is restricted to grid lines
Always greater than or equal to Euclidean distance
Works in any number of dimensions
Understanding Manhattan Distance
What is Manhattan Distance?
The Manhattan distance, also known as taxicab distance or city block distance, is a metric that measures the distance between two points by summing the absolute differences of their coordinates. Unlike Euclidean distance, which measures the straight-line distance, Manhattan distance represents the distance you would travel along a grid-like path.
Mathematical Formula
For two N-dimensional points a⃗ = [a₁, a₂, ..., aₙ] and b⃗ = [b₁, b₂, ..., bₙ]:
d = |a₁ - b₁| + |a₂ - b₂| + ... + |aₙ - bₙ|
Why "Manhattan"?
The name comes from the grid-like street layout of Manhattan, New York City. When walking from one point to another in Manhattan, you can't walk diagonally through buildings - you must follow the streets and avenues, which form a grid pattern. The total distance walked equals the Manhattan distance.
Practical Applications
Urban Navigation
GPS routing in cities with grid layouts, taxi fare calculations
Game Development
Pathfinding algorithms, movement in grid-based games
Data Science
Feature similarity, clustering algorithms, recommendation systems
Computer Vision
Image processing, pixel distance calculations
Key Properties
- •Always non-negative (d ≥ 0)
- •Symmetric: d(A,B) = d(B,A)
- •Satisfies triangle inequality
- •d(A,B) = 0 if and only if A = B