Case 1
- x₁ (first point)
- 0
- y₁ (first point)
- 0
- z₁ (first point)
- 0
- x₂ (second point)
- 2
- y₂ (second point)
- 3
- z₂ (second point)
- 6
7.000000
Open with these values3D Distance Calculator
7.000000
Result: 7.000000Subtract the coordinates axis by axis, square the three gaps, add them and take the square root. From the origin to (2, 3, 6) that is √(4 + 9 + 36) = 7. If both z values are equal the third term vanishes and the answer matches the plane formula.
The numbers at a glance
Held fixed: x₁ (first point) 0.0000, y₁ (first point) 0.0000, z₁ (first point) 0.0000, x₂ (second point) 2.0000, y₂ (second point) 3.0000.
| z₂ (second point) | Result |
|---|---|
| 0.0000 | 3.605551 |
| 2.0000 | 4.123106 |
| 4.0000 | 5.385165 |
| 6.0000Your value | 7.000000 |
| 8.0000 | 8.774964 |
| 10.0000 | 10.630146 |
| 12.0000 | 12.529964 |
Worked examples
Case 2
- x₁ (first point)
- 1
- y₁ (first point)
- 1
- z₁ (first point)
- 1
- x₂ (second point)
- 4
- y₂ (second point)
- 5
- z₂ (second point)
- 1
5.000000
Open with these valuesCase 3
- x₁ (first point)
- -1
- y₁ (first point)
- -2
- z₁ (first point)
- -2
- x₂ (second point)
- 2
- y₂ (second point)
- 2
- z₂ (second point)
- 4
7.810250
Open with these valuesHow it's calculated
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
- StepEnter the coordinates of the first point, then of the second.
- StepThe three axis gaps Δx, Δy and Δz are the edges of a box.
- ResultThe distance is its body diagonal: √(Δx² + Δy² + Δz²).
Reference table
| x₁, y₁, z₁, x₂, y₂, z₂ | Δx, Δy, Δz | Distance |
|---|---|---|
| 3, 3, 3, 3, 3, 3 | 0, 0, 0 | 0.000000 |
| 0, 0, 0, 1, 2, 2 | 1, 2, 2 | 3.000000 |
| 0, 0, 0, 1.5, 2.5, 3.5 | 1.5, 2.5, 3.5 | 4.555217 |
| 1, 1, 1, 4, 5, 1 | 3, 4, 0 | 5.000000 |
| 0, 0, 0, 2, 3, 6 | 2, 3, 6 | 7.000000 |
| -1, -2, -2, 2, 2, 4 | 3, 4, 6 | 7.810250 |
Questions
How do I calculate the distance between two points in 3D?
Use d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). Subtract the matching coordinates on each axis, square each difference, add the three squares and take the square root. For (0, 0, 0) and (1, 2, 2) that is √(1 + 4 + 4) = √9 = 3.
Why is this the Pythagorean theorem?
The two points sit at opposite corners of a box whose edges are the axis gaps Δx, Δy and Δz. Applying Pythagoras twice — first across the base, then up to the far corner — gives d² = Δx² + Δy² + Δz².
How is 3D distance different from 2D distance?
The 3D formula simply adds a third squared term for the z-axis. If both z-coordinates are equal, Δz is zero and the result matches the plane formula exactly.
Can I use negative or zero coordinates?
Yes, on any axis. For (−1, −2, −2) and (2, 2, 4) the gaps are 3, 4 and 6, giving √61 ≈ 7.81. Two identical points give a distance of zero.
Where is the 3D distance formula used?
Anywhere a straight-line gap in space matters: 3D modelling and CAD, collision checks in games, the separation between atoms in chemistry, robotics and motion planning.
Sources and last check
Information, not professional advice.