3D Distance Calculator: Find Distance Between Two Points in Space

What is the 3D Distance Between Two Points?

The 3D distance between two points is the length of the straight line connecting two locations in three-dimensional space. It's like measuring the direct path between two stars in the universe, considering not just left-right and up-down, but also forward-backward.

How to Calculate the 3D Distance Between Two Points

To find the 3D distance between two points, we use their x, y, and z coordinates. This method extends the familiar 2D distance formula to include the third dimension.

Formula

The formula for finding the distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D space is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Where:

• \(d\) is the 3D distance between the two points
• \(x_1, y_1, z_1\) are the coordinates of the first point
• \(x_2, y_2, z_2\) are the coordinates of the second point
• \(\sqrt{}\) denotes the square root operation

Calculation Steps

1. Identify the coordinates of your two points: (x₁, y₁, z₁) and (x₂, y₂, z₂)
2. Calculate the differences: (x₂ - x₁), (y₂ - y₁), and (z₂ - z₁)
3. Square each difference
4. Sum the squared differences
5. Take the square root of the sum

Example and Visual Representation

Let's calculate the distance between points A(1, 2, 3) and B(4, 6, 8):

Step 1: We have (x₁, y₁, z₁) = (1, 2, 3) and (x₂, y₂, z₂) = (4, 6, 8)

Step 2: x₂ - x₁ = 4 - 1 = 3, y₂ - y₁ = 6 - 2 = 4, z₂ - z₁ = 8 - 3 = 5

Step 3: (x₂ - x₁)² = 3² = 9, (y₂ - y₁)² = 4² = 16, (z₂ - z₁)² = 5² = 25

Step 4: 9 + 16 + 25 = 50

Step 5: \(\sqrt{50} \approx 7.07\)

Therefore, the distance between A and B is approximately 7.07 units.

This 3D representation shows points A and B in space. The red line illustrates the direct distance between them. Notice how the distance considers all three dimensions, creating a diagonal line that doesn't align with any single axis. This visual helps us understand why we need to account for changes in x, y, and z when calculating 3D distances.