Manhattan Distance Calculator

Manhattan distance measures how far apart two points are when movement occurs independently along coordinate axes. In two dimensions, imagine navigating a rectangular street grid: you cannot cut diagonally through buildings, so the route length is the horizontal separation plus the vertical separation. This gives the metric its names Manhattan distance, taxicab distance, and city-block distance. In linear algebra and data science, the same quantity is called the L1 norm of the difference between two vectors. For points P and Q, subtract corresponding coordinates, take the absolute value of each difference, and add those absolute differences. Absolute values matter because distance cannot be canceled by direction. From P at 1, 2 to Q at 4, 6, the coordinate changes have magnitudes three and four, so the Manhattan distance is seven. Any shortest axis-aligned route with those total movements has the same length, even though many different turn sequences are possible. The definition extends directly beyond a map. Three-dimensional points add the absolute difference in height or depth. A data record with ten numeric features can be treated as a point in ten-dimensional space, provided all features use comparable scales. The calculator therefore accepts any positive number of coordinates. Both points must have the same dimension because every coordinate in the first point needs a corresponding coordinate in the second. Manhattan distance is common in grid pathfinding, clustering, nearest-neighbor methods, image processing, warehouse routing, and robust statistics. Compared with Euclidean distance, it does not square large coordinate differences, so one unusually large deviation has less disproportionate influence. Its distance contours form diamonds in two dimensions rather than circles. Which metric is appropriate depends on the geometry of the problem: use Manhattan distance when movement is axis constrained or when additive feature differences model the intended cost. Enter coordinates separated by commas, spaces, or a mixture of both. Negative and decimal values are supported. The result includes the component expression so you can see exactly which absolute differences were added. Coordinate units carry through unchanged: inputs measured in blocks produce a result in blocks, while normalized feature inputs produce a unitless score. Before comparing real-world data, consider scaling variables with very different ranges, since a feature measured in thousands can otherwise dominate one measured between zero and one.