3D Distance Calculator

Math

This 3D distance calculator finds the straight-line distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in space. It extends the Pythagorean theorem to three dimensions: Distance = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. Enter the coordinates of both points to get the exact length of the segment that joins them.

Formula

Distance = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
  • This extends the 2D distance formula to space: it is the Pythagorean theorem applied across the x, y, and z differences.
  • Formula: Distance = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
  • Squaring each difference removes its sign, so the order of the two points does not change the result.
  • Use the same unit for all six coordinates; the distance comes out in that unit.
  • The distance is 0 only when both points coincide, and it is never negative.

Example Calculation

Inputs
  • X1: 0
  • Y1: 0
  • Z1: 0
  • X2: 3
  • Y2: 4
  • Z2: 12

From (0, 0, 0) to (3, 4, 12): the differences are 3, 4, and 12, so the distance is √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Frequently asked questions

What is the 3D distance formula?
It gives the straight-line distance between two points in space: Distance = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²], a direct extension of the Pythagorean theorem.
When should I use this calculator?
Use it whenever you have the (x, y, z) coordinates of two points and need the straight-line distance between them - in 3D geometry, physics, CAD, or 3D graphics.
Does the order of the points matter?
No. Each coordinate difference is squared, so swapping the points gives the same distance.
What units does the result use?
The same unit as the coordinates. If the coordinates are in metres, the distance is in metres.
Can the distance be negative?
No. A distance is a length, so it is always zero or positive - zero only when the two points are the same.