What is the 3D distance formula?

The distance between two points P1(x₁, y₁, z₁) and P2(x₂, y₂, z₂) is d = sqrt((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²). It extends the 2-D Pythagorean theorem into a third axis by adding the squared z-gap inside the square root.

How does the 3D formula relate to the Pythagorean theorem?

In 2-D the hypotenuse of a right triangle is sqrt(a² + b²). In 3-D you add a third perpendicular leg: sqrt(a² + b² + c²). You can think of it as two applications of Pythagoras — first combine x and y into a floor diagonal, then combine that diagonal with z to get the space diagonal.

What does the distance formula measure exactly?

It gives the straight-line (Euclidean) distance — the shortest possible path through open space between the two points. It does not account for obstacles, surfaces or curvature.

Why are there two solutions when solving for a coordinate?

If you fix the distance and five of the six coordinates, the unknown coordinate can sit on either side of the known point at the same distance — like two points on a circle. Both solutions satisfy d² = dx² + dy² + dz². Pick the one that fits your geometry.

Does this work with negative coordinates?

Yes. Each difference is squared before summing, so negative values and any order of the two points always give the same positive distance.

Can I use this for real-world 3D measurements?

Yes, for any Cartesian coordinate system — room dimensions in metres, game-world units, CAD model coordinates, GPS XYZ in ECEF, and so on. Just make sure all three axes use the same unit and the coordinates are in a flat (non-curved) reference frame.

3D Distance Calculator

The 3D distance calculator finds the exact straight-line (Euclidean) distance between any two points in three-dimensional space. Give it the (x, y, z) coordinates of both points and it instantly shows the distance, the full step-by-step working, and a small isometric diagram of the two points on the axes. You can also reverse the calculation: choose any one of the six coordinates as the unknown, supply the target distance, and the tool finds the two possible values for that coordinate.

This is the go-to tool for geometry homework, computer graphics, game development, CAD and engineering, robotics path planning, or any calculation that involves a gap through open 3-D space.

How it works

The 3-D distance formula is a direct generalisation of the 2-D Pythagorean theorem. Given two points P1(x₁, y₁, z₁) and P2(x₂, y₂, z₂), the differences along each axis are:

dx = x₂ − x₁, dy = y₂ − y₁, dz = z₂ − z₁

These three gaps form the legs of a right-angled box. The space diagonal — the straight-line distance — follows from squaring each leg, summing, and taking the square root:

d = sqrt(dx² + dy² + dz²)

Because every term is squared, the sign of each difference and the order of the two points never affect the result. The distance is always a non-negative number.

The solve-for mode inverts the formula. For example, if you know the distance d and want to find x₁, you rearrange to:

x₁ = x₂ ± sqrt(d² − dy² − dz²)

This has two solutions (the ± sign) because a sphere of radius d centred on P2 intersects the line parallel to the x-axis at two points. The calculator shows both; you choose whichever matches your context. If d² − dy² − dz² is negative, no real solution exists — the target distance is too small for the remaining two-axis gaps.

Worked example

Find the distance between P1(1, 2, 3) and P2(4, 6, 3):

dx = 4 − 1 = 3 dy = 6 − 2 = 4 dz = 3 − 3 = 0 d = sqrt(3² + 4² + 0²) = sqrt(9 + 16 + 0) = sqrt(25) = 5

The z-coordinates are equal, so this collapses to the classic 3-4-5 right triangle in the xy-plane.

Now a fully three-dimensional example — P1(0, 0, 0) and P2(1, 2, 2):

dx = 1, dy = 2, dz = 2 d = sqrt(1 + 4 + 4) = sqrt(9) = 3

P1	P2	Distance
(0, 0, 0)	(1, 2, 2)	3
(1, 2, 3)	(4, 6, 3)	5
(0, 0, 0)	(2, 4, 4)	6
(-1, -1, -1)	(2, 3, 3)	sqrt(41) approx 6.403

Formula note

The formula d = sqrt((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²) is the L2 norm (Euclidean norm) of the displacement vector from P1 to P2. It generalises to any number of dimensions: for n-dimensional space simply sum n squared differences under the square root. The 2-D version (no z term) is the standard school distance formula; the 3-D version adds one more squared difference to account for depth.