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Home Page > Math > Geometry Calculators

3D Distance Calculator

Calculate the Euclidean distance between two points in 3D space. Enter both sets of x, y, z coordinates to get the distance, midpoint, displacement vector and direction angles, with step-by-step formulas and a 3D diagram.

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Examples:
LIVE 3D PREVIEW
Point A (x₁, y₁, z₁)
to
Point B (x₂, y₂, z₂)

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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.

Real-World Applications

🎮
Game Development
Collision detection and pathfinding in 3D worlds
🤖
Robotics
Motion planning and obstacle avoidance
🏗
Architecture
Measuring diagonal spans in structures
🌌
Astronomy
Distances between stars and galaxies
📡
GPS & Navigation
3D positioning with altitude data
🧬
Molecular Biology
Atom-to-atom distances in proteins

Key Formulas

For two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) in 3D space:

PropertyFormulaDescription
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.

How to Use the 3D Distance Calculator

  1. 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.
  2. Enter Point B coordinates: Type the x₂, y₂, and z₂ values for the second point.
  3. Watch the live preview: The isometric 3D preview updates in real-time as you type, showing the spatial relationship between the two points.
  4. Click Calculate Distance: Press the button to compute all results.
  5. 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

What is the 3D distance formula?
The 3D distance formula is d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). It calculates the Euclidean (straight-line) distance between two points in three-dimensional space. It is derived by applying the Pythagorean theorem twice — once in the xy-plane and once incorporating the z-axis.
How is 3D distance different from 2D distance?
The 3D distance formula adds a third term (z₂ − z₁)² under the square root compared to the 2D formula. In 2D, you only consider the horizontal (x) and vertical (y) differences. In 3D, you also account for depth (z-axis), which measures how far apart the points are in the third dimension.
What are direction cosines in 3D?
Direction cosines are the cosines of the angles that the line segment between two points makes with the positive x, y, and z axes. They are calculated as cos(α) = Δx/d, cos(β) = Δy/d, and cos(γ) = Δz/d, where d is the distance. The sum of the squares of direction cosines always equals 1.
What is the midpoint formula in 3D?
The midpoint of two points in 3D space is M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). It gives the point that is exactly halfway between the two given points along each coordinate axis.
What is the difference between Euclidean and Manhattan distance?
Euclidean distance is the straight-line distance between two points — the shortest possible path through space. Manhattan distance (also called taxicab distance) is the sum of absolute differences along each axis, representing movement restricted to axis-aligned directions (like walking on a city grid). Manhattan distance is always greater than or equal to Euclidean distance.

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

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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