Distance Calculator (3D)

The 3D distance formula calculates the straight-line distance between two points in three-dimensional space: d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. It is an extension of the 2D Pythagorean theorem applied to a third axis (Z). The result is always a non-negative value.

Subtract the corresponding coordinates of the two points to get ΔX, ΔY, and ΔZ. Square each difference, add the three squared values together, then take the square root of the sum. For example, between (5, 6, 2) and (−7, 11, −13): d = √[(−12)² + 5² + (−15)²] = √[144 + 25 + 225] = √394 ≈ 19.849.

Using the 3D distance formula: d = √[(3−1)² + (6−1)² + (9−1)²] = √[4 + 25 + 64] = √93 ≈ 9.644. Enter these values directly into the calculator above to verify.

d = √[(0−1)² + (0−1)² + (0−1)²] = √[1 + 1 + 1] = √3 ≈ 1.732. This is also known as the magnitude of the vector (1, 1, 1).

Yes. The calculator accepts any real numbers — positive, negative, or decimal — for all six coordinate inputs. The squared differences in the formula ensure negative values are handled correctly and the result is always positive.

The 2D distance formula is d = √[(x₂−x₁)² + (y₂−y₁)²], which works on a flat plane. The 3D formula adds a third term for the Z-axis: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. Use the 3D formula whenever your points exist in three-dimensional space.

Distance is a scalar quantity representing a physical length, which cannot be negative. Even though the coordinate differences (ΔX, ΔY, ΔZ) can be negative, squaring them before summing ensures all contributions are positive, and the square root of a positive number is always positive.

The calculator is unit-agnostic — it works with whatever unit system your coordinates are in. If your coordinates are in meters, the result is in meters. If they are in inches, feet, or any other unit, the output distance will be in those same units.