# 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 The first point
X1
Y1
Z1
The second point
X2
Y2
Z2
The distance
Distance (d)
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