3D Distance Calculator

Finding the distance between two points in 3D space comes down to a single formula: the Pythagorean theorem with one more term tacked on for the z-axis. Plug in two sets of coordinates and this calculator returns the Euclidean distance, which is the shortest straight line between the two points. It works with positive numbers, negative numbers, and zeros in any combination, and it stays unit-agnostic so long as both points use the same unit on each axis.

How the formula works

The 3D distance formula is the same idea as the 2D version, just with a third squared difference added under the radical:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Subtract the matching coordinates for each axis, square each difference, add the three squares together, then take the square root. The squaring step is why the result is always non-negative; direction doesn't matter here, only the gap between the points.

A quick walkthrough: say the first point is (1, 2, 3) and the second is (4, 6, 8). The differences are 3, 4, and 5. Square them to get 9, 16, and 25. That sums to 50, and the square root of 50 is about 7.07. So the two points sit roughly 7.07 units apart.

Using the calculator

Type the X, Y, and Z values for point 1, then do the same for point 2. The distance shows up as soon as all six fields are filled. Order doesn't matter: swapping the two points gives the same result. If two points share a coordinate on one axis, the formula naturally collapses to the 2D distance on the remaining axes.

Where 3D distance shows up

Game engines lean on this formula constantly. Anytime a game checks whether a player is close enough to a checkpoint, an enemy, or a pickup, it's running some variant of the Euclidean distance under the hood (often the squared version, since skipping the square root is cheaper when you only need to compare).

Physics problems involving 3D displacement use it directly, as do robotics path planners deciding how far an end-effector has to travel. In machine learning it measures how close two feature vectors sit in 3D, which is the basis for nearest-neighbor classification and k-means clustering. GPS goes one step further by treating altitude as a third axis, so the same formula gives you a slant distance that accounts for elevation rather than just the flat ground distance.

Frequently asked questions

Can coordinates be negative?

Yes. The squaring step erases any sign, so (-5, 2, -3) to (1, -4, 7) works the same as any other pair. Mix positives, negatives, and zeros freely.

What units should I use?

Whatever you want. The formula has no opinion: enter meters and you get meters back, enter feet and you get feet. The only rule is consistency. Both points have to use the same unit on each axis. Mixing meters on one axis and feet on another will give you a number that doesn't mean anything.

Is this the same as Manhattan distance?

No. Manhattan distance sums the absolute differences instead of squaring them, which gives the distance along a grid (think driving across a city block plan). Euclidean is the straight-line shortest path through space, which is what most geometry, physics, and graphics problems actually want.

Why is the answer always positive?

Distance is a magnitude, not a vector. The squares throw away the sign of each axis difference, and the square root of a sum of non-negative numbers stays non-negative. If two points are identical, the distance is exactly zero.