How this instrument works
The 3D distance formula finds the straight-line distance between two points in space by extending the ordinary two-dimensional case one axis further: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Where a flat-plane calculation only needs a horizontal gap and a vertical gap, this one adds a third difference, the gap along the z-axis, squared and added under the same root alongside the other two. Nothing about the underlying logic changes — only the count of legs feeding into it grows from two to three.
This sheet is the direct upstream extension of the site's plain Distance Between Two Points calculator, which stops at two coordinates per point and has no z-axis field at all. Add a third coordinate to each point here and the same squared-difference-summed-under-a-root pattern simply keeps going: one more subtraction, one more square, one more term inside the sum before the root is taken. Anyone comfortable with the flat version already understands this one — the only new step is remembering to fill in z.
A useful check is what happens when the two points coincide: every one of the three differences vanishes, so the sum under the root is zero and the answer comes back as exactly 0, regardless of how many coordinates are involved. The same squared-and-summed pattern keeps extending cleanly beyond three dimensions too — scientists routinely measure the gap between points with dozens of coordinates using the identical structure, one more squared term per extra axis.
- Enter the first point's coordinates into Point 1: x, Point 1: y, and Point 1: z.
- Enter the second point's coordinates into Point 2: x, Point 2: y, and Point 2: z.
- Read Distance: the straight-line gap between the two points through all three axes.
- Negative coordinates and decimals are handled the same way — each difference is squared before anything else happens.
Worked example — from (0,0,0) to (3,4,12)
Take the points (0, 0, 0) and (3, 4, 12). The three differences are 3−0=3, 4−0=4, and 12−0=12. Squaring and summing gives 3² + 4² + 12² = 9 + 16 + 144 = 169, and the square root of 169 is exactly 13 — a clean whole-number result, a 3-4-12-13 Pythagorean quadruple sitting inside three-dimensional space.
Drop the z-coordinate entirely and only x and y survive: the same two points collapse to (0,0) and (3,4), whose flat gap is √(9+16) = √25 = 5, the familiar 3-4-5 triangle. Restoring the z-difference of 12 stretches that 5-unit gap into the full 13-unit line through space — the extra axis adds real length, it doesn't just pad the arithmetic.
Questions
What is the 3D distance formula?
d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²), where each point carries three coordinates. It's the ordinary two-point formula with one extra squared term added for the z-axis, still built entirely from the Pythagorean theorem.
How does this differ from the 2D distance calculator?
It adds a third coordinate, z, to each point and a matching (z₂−z₁)² term under the root. The site's plain Distance Between Two Points sheet only ever handles flat, two-coordinate points; this one is for locations in actual three-dimensional space, such as positions in a room or a 3D model.
What is the gap between two identical points in 3D?
Exactly zero. All three coordinate differences vanish when the points coincide, leaving nothing under the root — the same outcome the flat version gives whenever its two points match.
Does the order I enter the two points matter?
No — each difference is squared before it's added to the total, and squaring erases whichever sign a swapped subtraction would introduce. Point 1 and Point 2 can be entered in either order with no change to the resulting length.
Can the same idea extend beyond three axes?
Yes — the pattern of squaring each coordinate's difference and summing them under a root keeps working for four coordinates, ten, or any number, a generalization mathematicians call Euclidean distance in n-dimensional space. Three axes simply match physical space directly.