To calculate the distance between two points in three-dimensional (3D) space, we utilize the distance formula, which is derived from Pythagoras’ theorem.

The distance formula for two points in 3D is an extension of the familiar 2D distance formula. Let’s denote two points as $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$. The distance $d$ between these two points can be calculated using the following formula:

This formula works by first determining the differences in the x, y, and z coordinates of the points. Each difference is squared, summed together, and finally, the square root of that sum is taken. This approach is a direct application of Pythagoras’ theorem extended into three dimensions.

For example, consider the points $A(1, 2, 3)$ and $B(4, 6, 8)$. We begin by calculating the differences in each coordinate:

Next, we square each of these differences:

Now, we sum these squared differences:

Finally, we take the square root of this sum to find the distance:

Therefore, the distance between points $A$ and $B$ is approximately $7.07$ units. This method allows you to accurately measure the straight-line distance between any two points in 3D space.