Distance Calculator

Calculate Euclidean, Manhattan, and Chebyshev distances between two points in 2D space with step-by-step solutions.

Enter Two Points

Point 1 (x1, y1)

Point 2 (x2, y2)

Result

Euclidean Distance
5
units
Euclidean Distance5
Manhattan Distance7
Chebyshev Distance4
Horizontal Distance (|dx|)3
Vertical Distance (|dy|)4

Step-by-Step Solution

d = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5

Understanding Distance Metrics

Distance is a numerical measure of how far apart two points are in a given space. While the Euclidean distance (straight-line distance) is the most familiar, there are several other distance metrics that are useful in different contexts. This calculator computes three common distance metrics between two points in 2D space.

Distance Formulas

Euclidean Distance (L2)

The straight-line "as the crow flies" distance. Derived from the Pythagorean theorem.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

Manhattan Distance (L1)

Also called taxicab or city-block distance. Sum of absolute differences along each axis.

d = |x2-x1| + |y2-y1|

Chebyshev Distance (L-infinity)

Maximum of the absolute differences. Equivalent to the minimum number of king moves in chess.

d = max(|x2-x1|, |y2-y1|)

Euclidean Distance In Depth

The Euclidean distance is the most commonly used distance metric. It represents the length of the shortest path between two points, which is a straight line. The formula is derived directly from the Pythagorean theorem: if you draw a right triangle with the two points as endpoints of the hypotenuse, the horizontal leg has length |x2 - x1| and the vertical leg has length |y2 - y1|. The hypotenuse (the distance) is then sqrt((x2-x1)2 + (y2-y1)2).

In higher dimensions, the formula generalizes naturally: d = sqrt((x2-x1)2 + (y2-y1)2 + (z2-z1)2 + ...) for as many dimensions as needed.

Manhattan Distance In Depth

The Manhattan distance gets its name from the grid-like street layout of Manhattan, New York City. If you can only travel along horizontal and vertical paths (like on city streets), the shortest path between two points is the sum of the horizontal and vertical distances. This metric is widely used in city planning, robotics (grid-based navigation), and as a heuristic in pathfinding algorithms like A*.

Chebyshev Distance In Depth

The Chebyshev distance (also called the chessboard distance) represents the minimum number of moves a king would need to travel between two squares on a chessboard. Since a king can move one square in any direction (including diagonally), the distance is simply the larger of the horizontal and vertical distances. This metric is useful in discrete mathematics, game theory, and warehouse logistics.

Relationship Between Distance Metrics

For any two points, these three distances satisfy the following inequalities:

  • Chebyshev distance is always less than or equal to Euclidean distance.
  • Euclidean distance is always less than or equal to Manhattan distance.
  • Manhattan distance is at most sqrt(2) times the Euclidean distance (in 2D).
  • Chebyshev <= Euclidean <= Manhattan

Applications

  • Euclidean: Physics, engineering, everyday measurements, GPS navigation.
  • Manhattan: Urban planning, VLSI design, warehouse routing, taxicab problems.
  • Chebyshev: Chess programming, industrial robotics, image processing.
  • Machine learning: All three metrics are used in k-nearest neighbors, clustering, and similarity measures.