What is the formula for the distance between two points?

In 2D: d = √[(x₂−x₁)² + (y₂−y₁)²]. In 3D add the z-term: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. It is the Pythagorean theorem applied once (2D) or twice (3D). Example: (0,0) to (3,4) gives √(9+16) = 5.

What is the difference between Euclidean and Manhattan distance?

Euclidean distance is the straight-line distance between two points. Manhattan distance (|dx| + |dy|) is the shortest path when you can only move along a grid, like walking city blocks. Manhattan distance is always greater than or equal to Euclidean distance — for (0,0) to (3,4): 7 vs 5.

What is Chebyshev distance?

Chebyshev distance is max(|dx|, |dy|): the number of moves a chess king needs to go from one square to another, since a king can move diagonally. It is always less than or equal to Euclidean distance — for (0,0) to (3,4): 4 vs 5.

How do I convert Minecraft Nether coordinates to Overworld coordinates?

Multiply the X and Z coordinates by 8; the Y (height) coordinate stays the same. Example: Nether (100, 64, 800) → Overworld (800, 64, 6400). To convert Overworld → Nether, divide X and Z by 8 instead.

How do you measure distance in Minecraft blocks?

Use the same Euclidean formula on the block coordinates — one block equals one meter. For long Overworld trips, divide the result by 8 to get the Nether-travel shortcut distance. Example: 1131.37 Overworld blocks ≈ 141.42 Nether blocks.

Can the distance between two points be negative or zero?

Never negative: it is the square root of a sum of squares, which is always ≥ 0. It is exactly zero only when both points are identical, e.g. (7,7) to (7,7).

When should I use 3D distance instead of 2D?

Use 3D whenever height or depth matters: Minecraft builds with elevation changes, drone flight paths, or the space diagonal of a room. If both points share the same z-value, the 3D formula gives exactly the same answer as 2D.

Is this the same as latitude/longitude distance?

No. Latitude and longitude lie on the curved surface of the Earth, so they need the haversine great-circle formula — see our latitude-longitude distance calculator. This tool is for flat Cartesian (x, y) or (x, y, z) coordinates, where the Pythagorean formula is exact.

NR Tech Studio

Distance Between Two Points Calculator

Compute the Euclidean (straight-line), Manhattan (grid) and Chebyshev distance between two points in 2D or 3D — with the full step-by-step formula, a live visualization, unit conversion, and a dedicated Minecraft mode (block distance plus Nether ↔ Overworld coordinate conversion at the 8:1 ratio).

2D distance — (x₁, y₁) to (x₂, y₂)

x₁ y₁ x₂ y₂

Unit

Worked examples

Example 1 — classic 3-4-5 triangle (2D).
Points (0, 0) → (3, 4): dx = 3, dy = 4.
d = √(3² + 4²) = √(9 + 16) = √25 = 5.
Manhattan = 3 + 4 = 7; Chebyshev = max(3, 4) = 4.

Example 2 — 3D with height.
Points (1, 2, 3) → (4, 6, 8): dx = 3, dy = 4, dz = 5.
d = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 ≈ 7.0711.
Manhattan = 3 + 4 + 5 = 12.

Example 3 — negative coordinates (2D).
Points (−2, −3) → (4, 5): dx = 6, dy = 8.
d = √(6² + 8²) = √(36 + 64) = √100 = 10.
Manhattan = 6 + 8 = 14; Chebyshev = max(6, 8) = 8.

Example 4 — Minecraft Nether travel.
Nether portal at (100, 800) → Overworld (100×8, 800×8) = (800, 6400) (Y unchanged).
Distance in the Overworld from (800, 6400) to (1600, 7200): dx = 800, dz = 800.
d = √(800² + 800²) = √1,280,000 ≈ 1131.3708 blocks (≈ 1131.37 m).
Nether-travel equivalent: 1131.3708 ÷ 8 ≈ 141.4214 blocks; Manhattan = 1600 blocks.

Scope boundary. This tool covers flat Cartesian distance (2D/3D) plus Minecraft block math, and absorbs the planned coordinate-distance tool (#1127 — killed pre-build as identical intent). It does not do great-circle distance on Earth's curved surface (latitude/longitude need the haversine formula), item-stack math, or sound falloff — see the related tools below.

Related tools

Latitude–Longitude Distance — great-circle (haversine) distance between GPS coordinates.
Minecraft Stack Calculator — items ↔ stacks, shulker boxes and storage planner.
Distance Attenuation — how sound level drops with distance.

Frequently asked questions

Free tool by NRTechStudio

Distance Between Two Points Calculator

2D distance — (x₁, y₁) to (x₂, y₂)

Step by step

Visualization

3D distance — (x₁, y₁, z₁) to (x₂, y₂, z₂)

Step by step

Minecraft mode — blocks & Nether ↔ Overworld

Visualization

Worked examples

Frequently asked questions

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