Calculate the distance between two points in 2D or 3D using Euclidean, Manhattan, Chebyshev, and Minkowski metrics. Bearing, direction angle, component breakdown, and distance metrics reference.
| Component | |Δ| | Δ² | % of Euclidean² |
|---|---|---|---|
| Δx | 0.0000 | 0.0000 | 0.0% |
| Δy | 0.0000 | 0.0000 | 0.0% |
| Total | — | 0.0000 | 100% |
| Metric | Formula | Also Called | Use Case |
|---|---|---|---|
| Euclidean (L²) | √(Σ(xᵢ−yᵢ)²) | Straight-line, Pythagorean | Geometry, physics, ML clustering |
| Manhattan (L¹) | Σ|xᵢ−yᵢ| | Taxicab, city-block | Grid navigation, sparse data |
| Chebyshev (L∞) | max|xᵢ−yᵢ| | Chessboard | Chess king moves, logistics |
| Minkowski (Lᵖ) | (Σ|xᵢ−yᵢ|ᵖ)^(1/p) | Generalized norm | Generalization of above metrics |
| Squared Euclidean | Σ(xᵢ−yᵢ)² | SSD | Comparisons (no square root) |
| Haversine | Great-circle formula | Spherical distance | GPS / geographic coordinates |
The distance formula calculates how far apart two points are in a coordinate system. In its most familiar form — the Euclidean distance — it is the straight-line distance between two points derived from the Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²). This extends to three dimensions by adding a z-component under the radical.
But Euclidean distance is not the only way to measure "how far." In a grid-based city, the Manhattan (taxicab) distance — the sum of absolute differences along each axis — better represents travel distance when you must follow streets. In chess, the Chebyshev distance — the maximum absolute difference on any axis — tells you how many moves a king needs. The Minkowski distance generalizes all of these with a parameter p: p=1 gives Manhattan, p=2 gives Euclidean, and p→∞ gives Chebyshev.
This calculator computes all four metrics simultaneously so you can compare them side by side. It also provides the squared Euclidean distance (useful in machine learning where the square root is unnecessary for comparisons), the direction angle from the positive x-axis, the compass bearing, and a detailed component breakdown showing how much each axis contributes to the total distance. Whether you are a student learning coordinate geometry, a programmer implementing nearest-neighbor search, a game developer computing tile distances, or an analyst choosing the right metric for clustering, the page keeps those measures attached to the same pair of points.
The Distance Formula Calculator — Euclidean, Manhattan, Chebyshev & More is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Euclidean Distance, Euclidean² (Squared), Manhattan Distance in one pass, with conversions and derived values shown together.
Euclidean: d = √((x₂−x₁)² + (y₂−y₁)² [+ (z₂−z₁)²])
Manhattan: d = |x₂−x₁| + |y₂−y₁| [+ |z₂−z₁|]
Chebyshev: d = max(|x₂−x₁|, |y₂−y₁| [, |z₂−z₁|])
Minkowski(p): d = (|Δx|ᵖ + |Δy|ᵖ [+ |Δz|ᵖ])^(1/p)
Direction: θ = atan2(Δy, Δx)
Bearing: (90° − θ + 360°) mod 360°Result: Euclidean = 5, Manhattan = 7, Chebyshev = 4
Euclidean: √(3² + 4²) = √(9+16) = √25 = 5 (the classic 3-4-5 right triangle). Manhattan: |3| + |4| = 7 (walk 3 blocks east, then 4 blocks north). Chebyshev: max(3, 4) = 4 (a chess king could reach (3,4) in 4 moves). Manhattan/Euclidean ratio = 7/5 = 1.4.
Distance Formula Calculator — Euclidean, Manhattan, Chebyshev & More calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
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The distance formula calculates the straight-line (Euclidean) distance between two points: d = √((x₂−x₁)² + (y₂−y₁)²). It is derived from the Pythagorean theorem.
Manhattan (taxicab) distance is the sum of the absolute differences along each axis: |Δx| + |Δy|. It represents the distance traveled on a rectangular grid.
Chebyshev distance is the maximum of the absolute differences along any axis: max(|Δx|, |Δy|). It is the number of moves a chess king needs to travel between two squares.
Use Manhattan when movement is restricted to a grid (city blocks, pixel grids). Use Euclidean when straight-line movement is possible (flying, open terrain).
Yes. Switch to 3D mode and enter Z-coordinates. All distance metrics extend naturally to three (or more) dimensions.
Direction angle is measured counter-clockwise from the positive x-axis (math convention). Compass bearing is measured clockwise from north (navigation convention).
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