3D Distance Calculator
MiniWebtool’s 3D Distance Calculator finds the straight-line distance between two spatial points from their x, y, and z coordinates. A 3D diagram and worked formulas connect the distance with midpoint and displacement results.
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Enter the x, y, z coordinates of two points below to get the distance, the midpoint, the displacement vector and the direction angles, with the steps.
About 3D Distance Calculator
The 3D Distance Calculator computes the Euclidean distance between two points in three-dimensional space using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). Enter the coordinates of Point A \((x_1, y_1, z_1)\) and Point B \((x_2, y_2, z_2)\) to instantly get the distance, midpoint, displacement vector, direction angles, and alternative distance metrics (Manhattan and Chebyshev) with step-by-step formulas and an interactive 3D diagram.
At a Glance
3D Distance Calculator finds the straight-line distance between two points from their x, y and z coordinates. It also returns the midpoint, displacement vector, direction angles and other distance metrics.
| Feature | What you get | Good for |
|---|---|---|
| Two-point input | Enter x, y and z for Point A and Point B. Preset examples fill both points: unit diagonal, architecture, game world, star map and physics. | Trying the tool with a ready-made case |
| Distance formula | d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²), with step-by-step working. | Showing the Pythagorean theorem extended to 3D |
| Midpoint and displacement | The point halfway between A and B, plus the displacement vector from A to B. | Locating a center point or a movement vector |
| Direction angles | Direction cosines give the angles the segment makes with the x, y and z axes. | Orientation work in physics, engineering and graphics |
| Other metrics | Manhattan distance (sum of axis differences) and Chebyshev distance (largest axis difference) are shown beside the Euclidean result. | Grid-style movement and worst-axis comparisons |
| 3D diagram | A live preview updates as you type. Toggle axes, projections, midpoint and an xy-plane grid. | Visualizing where two points sit in space |
Real-World Applications
Key Formulas
For two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) in 3D space:
| Property | Formula | Description |
|---|---|---|
| Euclidean Distance | \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}\) | Straight-line distance through space |
| Midpoint | \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\) | Point exactly halfway between A and B |
| Manhattan Distance | \(d_M = |\Delta x| + |\Delta y| + |\Delta z|\) | Sum of axis-aligned distances |
| Chebyshev Distance | \(d_C = \max(|\Delta x|, |\Delta y|, |\Delta z|)\) | Maximum difference along any axis |
| Direction Cosines | \(\cos\alpha = \frac{\Delta x}{d}\) \(\cos\beta = \frac{\Delta y}{d}\) \(\cos\gamma = \frac{\Delta z}{d}\) | Angles with the coordinate axes |
Understanding the 3D Distance Formula
The 3D distance formula is an extension of the Pythagorean theorem. In 2D, the distance between two points is \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2}\). To extend this to 3D, we apply the theorem twice: first in the xy-plane to get the horizontal distance, then combine that with the z-difference. The result is \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}\). This formula gives the length of the shortest path (a straight line) between two points in Euclidean space.
Units and Input Format
The calculator does not have a unit selector. The results are unitless, so the distance is in the same unit you used for the coordinates. For example, if you enter coordinates in meters, the distance is in meters. Spaces are ignored. Numbers with a comma followed by exactly three digits (such as 1,000) are read as thousands separators, while other commas are read as decimal points.
How to Use the 3D Distance Calculator
- Enter Point A coordinates: Type the x₁, y₁, and z₁ values for the first point, or click a quick example to auto-fill both points.
- Enter Point B coordinates: Type the x₂, y₂, and z₂ values for the second point.
- Watch the live preview: The isometric 3D preview updates in real-time as you type, showing the spatial relationship between the two points.
- Click Calculate Distance: Press the button to compute all results.
- Review results: See the Euclidean distance, midpoint, displacement vector, direction angles, and alternative distance metrics. Toggle the diagram layers to visualize axes, projections, midpoint, and the xy-plane grid.
Euclidean vs. Manhattan vs. Chebyshev Distance
Euclidean distance is the straight-line distance — the shortest path through space. Manhattan distance (also called taxicab or L₁ distance) sums the absolute differences along each axis, like walking along a city grid where diagonal shortcuts are not allowed. Chebyshev distance (L∞ distance) is the maximum absolute difference along any single axis — it represents how far apart the points are in the "worst-case" dimension. Euclidean distance is always ≤ Manhattan distance, and Chebyshev distance is always ≤ Euclidean distance.
Direction Cosines and Angles
Direction cosines describe the orientation of the line segment from A to B relative to the coordinate axes. If \(\alpha\), \(\beta\), and \(\gamma\) are the angles the line makes with the x-, y-, and z-axes respectively, then \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\). This identity always holds and is a useful check for computation accuracy. Direction cosines are widely used in physics, engineering, and computer graphics for specifying orientations in 3D space.
FAQ
Reference this content, page, or tool as:
"3D Distance Calculator" at https://MiniWebtool.com/3d-distance-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-03
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