# Three-Dimensional Distance Calculator

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 = √[(x _{2} - x_{1})² + (y_{2} - y_{1})² +(z_{2} - z_{1})²]**

where:

**D**is the distance between two points;**(x**are the coordinates of the first point; and_{1}, y_{1}, z_{1})**(x**are the coordinates of the second point._{2}, y_{2}, z_{2})

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?

- Write down the coordinates of the first point:
**(x**._{1}, y_{1}, z_{1}) - Choose the coordinates of the second point:
**(x**._{2}, y_{2}, z_{2}) - Compute the differences between the corresponding coordinates:
**x**,_{2}- x_{1}**y**, and_{2}- y_{1}**z**._{2}- z_{1} **Square**the three values obtained in Step 3.**Add together**the three values obtained in Step 4.- Determine the
**square root**of the result. This is the answer!

### 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 = √[(x _{2} - x_{1})² + (y_{2} - y_{1})² +(z_{2} - z_{1})²]**. Clearly, we get

**√[(3-1)**, as claimed.

^{2}+ (6-1)^{2}+(9-1)^{2}] = √[4 + 25 + 64] = √93 = 9.643### 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**.