Finding the distance between two points in three-dimensional space comes up constantly in geometry, physics, computer graphics, and engineering — anywhere positions are tracked with x, y, and z coordinates. This calculator applies the 3D distance formula directly, and also surfaces two closely related results: the midpoint of the segment and the displacement vector between the points.
How the 3D distance formula works
The formula d = √(Δx²+Δy²+Δz²) is a direct extension of the 2D distance formula, which is itself the Pythagorean theorem in disguise. In 2D, you find the hypotenuse of a right triangle formed by Δx and Δy. In 3D, you apply the Pythagorean theorem twice: first to find the diagonal distance across the xy-plane (√(Δx²+Δy²)), then treat that diagonal and Δz as the two legs of a second right triangle rising into the z-axis. Squaring both sides and simplifying gives the compact three-term formula used here.
Inputs and what they mean
The calculator takes six numbers: the x, y, and z coordinates of Point A, and the x, y, and z coordinates of Point B. There are no required units — the formula works identically whether your coordinates are in meters, inches, pixels, or any other consistent unit, and the result comes out in that same unit. Negative coordinates are fully supported, since 3D space extends in both directions along every axis.
Limits and edge cases
When Point A and Point B are the same coordinate, the distance is 0 and the displacement vector is (0, 0, 0) — there's no meaningful direction to report. This formula measures straight-line ('as the crow flies') distance only; it does not account for obstacles, curved paths, or any routing constraint. It's also a different metric from Manhattan distance (which sums absolute coordinate differences instead of taking a square root of squares) — the two give different answers except along a single axis.