3D Distance Between Two Points
Take the planar Pythagoras and add one more squared term for the z difference. Square each axis difference, add, take the root. That the form survives the extra dimension is the power of the theorem.
Your numbers
Distance
14.1421
Result
10
Difference
10
Formula
Distance = √((x₂ − x₁) ² + (y₂ − y₁) ² + (z₂ − z₁) ²)
The horizontal run and vertical rise are shown alongside. For pipe or cable runs, add slack on top.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| x₁The horizontal coordinate of the first point. | 0 | No limit |
| y₁The vertical coordinate of the first point. | 0 | No limit |
| z₁The height coordinate of the first point. | 0 | No limit |
| x₂The horizontal coordinate of the second point. | 6 | No limit |
| y₂The vertical coordinate of the second point. | 8 | No limit |
| z₂The height coordinate of the second point. | 10 | No limit |
Step by step
Quick reference table
Results when only x₂ changes and everything else stays put.
| x₂ | Distance | Result | Difference |
|---|---|---|---|
| 3 | 13.1529 | 8.544 | 10 |
| 4.5 | 13.5739 | 9.1788 | 10 |
| 6 | 14.1421 | 10 | 10 |
| 9 | 15.6525 | 12.0416 | 10 |
| 12 | 17.5499 | 14.4222 | 10 |
What each result means
| Result | At default values |
|---|---|
| DistanceThe straight-line distance between two points. | 14.1421 |
| ResultWhat the formula gives back — the answer this page is for. | 10 |
| DifferenceThe gap between two values — a quantity, not a rate. | 10 |
Common mistakes
The horizontal run and vertical rise are shown alongside. For pipe or cable runs, add slack on top.
Glossary
- x₁
- The horizontal coordinate of the first point.
- y₁
- The vertical coordinate of the first point.
- z₁
- The height coordinate of the first point.
- x₂
- The horizontal coordinate of the second point.
- y₂
- The vertical coordinate of the second point.
- z₂
- The height coordinate of the second point.
- Distance
- The straight-line distance between two points.
- Result
- What the formula gives back — the answer this page is for.
- Difference
- The gap between two values — a quantity, not a rate.
Frequently asked questions
Q. How is 3D Distance Between Two Points calculated?
Distance = √((x₂ − x₁) ² + (y₂ − y₁) ² + (z₂ − z₁) ²) — Take the planar Pythagoras and add one more squared term for the z difference. Square each axis difference, add, take the root. That the form survives the extra dimension is the power of the theorem.
Q. Can you walk through an example?
With x₁ 0, y₁ 0, z₁ 0, x₂ 6, y₂ 8, z₂ 10, the answer is Distance 14.1421.
Q. What do I need to enter?
Enter x₁, y₁, z₁, x₂, y₂, z₂. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only x₂ moves the answer to x₂ 3 → Distance 13.1529 and x₂ 12 → Distance 17.5499. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The horizontal run and vertical rise are shown alongside. For pipe or cable runs, add slack on top.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.