Three-Dimensional Distance Calculator

Created by Bogna Szyk
Reviewed by Jack Bowater
Last updated: Jan 19, 2023

Our three-dimensional distance calculator is a tool that finds the distance between two points, provided you give their coordinates in space. If you want to determine the distance between two points on a plane (two-dimensional distance), use our distance calculator. You can also check our average rate of change calculator to determine the relation between two coordinates.

What is the 3D distance formula?

To find the distance between two points in a three-dimensional coordinate system, you need to apply the following formula:

D = √[(x2 - x1)² + (y2 - y1)² +(z2 - z1)²]

where:

  • D is the distance between two points;
  • (x1, y1, z1) are the coordinates of the first point; and
  • (x2, y2, z2) are the coordinates of the second point.

Notice that the value obtained when using this formula is always positive. This is because we consider a scalar value of distance - that is, it is impossible to have a negative value for distance. You can learn about the similarity between the distance formula and vector magnitude in our vector magnitude calculator.

How do I calculate the 3D distance?

  1. Write down the coordinates of the first point: (x1, y1, z1).
  2. Choose the coordinates of the second point: (x2, y2, z2).
  3. Compute the differences between the corresponding coordinates: x2 - x1, y2 - y1, and z2 - z1.
  4. Square the three values obtained in Step 3.
  5. Add together the three values obtained in Step 4.
  6. Determine the square root of the result. This is the answer!

FAQ

What is the distance from (1,1,1) to (3,6,9)?

The distance is 9.643. To get this answer, fill in the formula: D = √[(x2 - x1)² + (y2 - y1)² +(z2 - z1)²]. Clearly, we get √[(3-1)2 + (6-1)2 +(9-1)2] = √[4 + 25 + 64] = √93 = 9.643, as claimed.

What is the distance from (1,1,1) to the origin of the space?

The distance is √3, so around 1.73. To arrive at this result, you need to recall that the coordinates of the origin are (0, 0, 0). The 3D distance formula gives √[(1-0)2 + (1-0)2 +(1-0)2] = √[1 + 1 + 1] = √3.

Bogna Szyk
3D scatter plot with two points connected with a line representing the distance between them
The first point
X1
Y1
Z1
The second point
X2
Y2
Z2
The distance
Distance (d)
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