Distance Calculator · Practical guide
Distance Between Two Points in 3D
Add one more squared term for the z-axis — the 2D formula extends naturally into three dimensions.
At a glance
The 3D Distance Formula
For points (x₁, y₁, z₁) and (x₂, y₂, z₂), the distance is d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). It is the 2D formula with one extra squared difference for the vertical (z) axis — the Pythagorean theorem applied twice in a row.
Interactive tool
Distance Calculator
Enter your own values, then compare the result with the examples below.
Results
Worked Example
Find the distance from (1, 2, 3) to (4, 6, 8). The differences are 3, 4, and 5. Squared: 9, 16, and 25. Sum: 50. So d = √50 ≈ 7.0711. The midpoint — the center of the segment — is ((1+4)/2, (2+6)/2, (3+8)/2) = (2.5, 4, 5.5). The calculator above reports both values plus the step-by-step working when you use the 3D tab.
More 3D Practice
| Point 1 | Point 2 | Distance | Midpoint |
|---|---|---|---|
| (0, 0, 0) | (2, 3, 6) | 7 | (1, 1.5, 3) |
| (1, 1, 1) | (5, 5, 5) | ≈ 6.9282 | (3, 3, 3) |
| (−1, 2, 0) | (3, −2, 4) | ≈ 6.9282 | (1, 0, 2) |
Where 3D Distance Shows Up
- Computer graphics and games: every collision check and camera calculation measures 3D distances between objects.
- Physics and engineering: displacement between two positions in space, antenna spacing, and cable lengths.
- Data science: Euclidean distance between points with three features — the same formula extends to any number of dimensions.
Frequently Asked Questions
Why does the 3D formula have three squared terms?
Each axis contributes one leg of the right-triangle structure. Squaring and summing all three differences is equivalent to applying the Pythagorean theorem first in a plane, then again with the z-axis.
What is the midpoint formula in 3D?
Average each coordinate pair: ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). It is the point exactly halfway along the segment connecting the two endpoints.
Can I use negative coordinates in 3D?
Yes. Points in any octant work the same way because the differences are squared. From (−1, 2, 0) to (3, −2, 4) the distance is √48 ≈ 6.9282.
Is 3D distance ever zero or negative?
Distance is zero only when all three coordinates match — the same point — and it can never be negative, since it is the square root of a sum of squares.