3D Distance Calculator

What is the 3D Distance Between Two Points?

The 3D distance between two points in three-dimensional space ($A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$) is the length of the straight line segment connecting them. This three-dimensional Euclidean distance generalizes the classical 2D Pythagorean theorem to spatial coordinates.

The 3D Distance Formula

By treating the displacement in three orthogonal dimensions ($\Delta x, \Delta y, \Delta z$) as the edges of a rectangular prism (box), the distance between opposite diagonal vertices gives the 3D distance formula:

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

Where:

$d$ is the straight-line 3D Euclidean distance.
$x_1, y_1, z_1$ are the coordinates of the initial point $A$.
$x_2, y_2, z_2$ are the coordinates of the target point $B$.
$\Delta x = x_2 - x_1$ is the displacement along the X-axis.
$\Delta y = y_2 - y_1$ is the displacement along the Y-axis.
$\Delta z = z_2 - z_1$ is the displacement along the Z-axis.

Other 3D Distance Metrics

1. 3D Manhattan (Taxicab) Distance

Manhattan distance in 3D is the sum of absolute differences along all three coordinate axes:

$$d_{\text{Manhattan}} = |x_2 - x_1| + |y_2 - y_1| + |z_2 - z_1|$$

2. Chebyshev (Chessboard) Distance

Chebyshev distance measures the maximum coordinate difference along any single dimension:

$$d_{\text{Chebyshev}} = \max(|x_2 - x_1|, |y_2 - y_1|, |z_2 - z_1|)$$

3. 3D Midpoint Formula

The midpoint is the coordinate located precisely halfway along the line segment connecting both points:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$$

Step-by-Step Example Calculation

Let's calculate the distance between point $A(1, 2, 3)$ and point $B(4, 6, 8)$:

Find differences: $\Delta x = 4 - 1 = 3$, $\Delta y = 6 - 2 = 4$, $\Delta z = 8 - 3 = 5$.
Square the differences: $3^2 = 9$, $4^2 = 16$, $5^2 = 25$.
Sum squares: $9 + 16 + 25 = 50$.
Compute square root: $d = \sqrt{50} \approx 7.0711$.

You can also use our 2D Distance Calculator, analyze with the Distance Calculator, or try our Speed Distance Time Calculator.

Frequently Asked Questions

How does 3D distance differ from 2D distance?

In 2D distance calculations, only horizontal ($\Delta x$) and vertical ($\Delta y$) displacements are evaluated. In 3D space, depth ($\Delta z$) is added into the quadratic sum, extending the Pythagorean theorem to three dimensions.

What is the displacement vector between two points?

The displacement vector $\vec{AB} = (\Delta x, \Delta y, \Delta z)$ specifies the directional magnitude required to travel directly from point $A$ to point $B$. Its magnitude $|\vec{AB}|$ is the Euclidean distance.

Can 3D distance be used for physics and computer graphics?

Yes. 3D distance is foundational across 3D game engines, collision detection, CAD modeling, astrophysics, molecular modeling, and robotics navigation.

Does point order change the 3D distance?

No. Because $(x_2 - x_1)^2 = (x_1 - x_2)^2$, distance is symmetric ($d(A, B) = d(B, A)$).

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