3D Distance Calculator

Calculate the Euclidean distance between two points in three-dimensional space using the extended distance formula.

Enter 3D Coordinates

x y z P1(1,2,3) P2(4,6,8) M(2.5,4,5.5)
Point 1 (x₁, y₁, z₁)
Point 2 (x₂, y₂, z₂)

Result

Enter coordinates and click Calculate.

Understanding 3D Distance

The 3D Distance Formula

The distance between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) in three-dimensional Cartesian space is:

d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

This is a natural extension of the 2D distance formula into three dimensions, adding a third squared-difference term for the z-axis.

Extension from 2D to 3D

In two dimensions, the distance formula uses a single application of the Pythagorean theorem. In three dimensions, the theorem is applied twice: first in the xy-plane to get dxy = √[(Δx)² + (Δy)²], then treating dxy and Δz as the legs of a new right triangle to get the full 3D distance.

3D Coordinate Systems

The Cartesian coordinate system in 3D uses three mutually perpendicular axes (x, y, z). Other 3D coordinate systems include:

  • Cylindrical coordinates: (r, θ, z) where r is the distance from the z-axis.
  • Spherical coordinates: (ρ, θ, φ) where ρ is the distance from the origin.

The 3D Midpoint

M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)

Applications

  • Computer Graphics and Gaming: Collision detection and physics simulations in 3D environments.
  • Robotics: Measuring distances in 3D workspace for end-effector positioning.
  • Molecular Chemistry: Computing bond lengths and inter-atomic distances.
  • GPS and Surveying: Determining distances with elevation differences.
  • Astronomy: Measuring distances between celestial bodies.
  • 3D Printing: Calculating tool-path distances and material requirements.