Distance Formula Calculator

About Distance Formula Calculator

Distance formula derives from the Pythagorean theorem. In 3D: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²).

How to use this calculator

1. Enter the coordinates of the first point.
2. Enter the coordinates of the second point.
3. Choose 2D or 3D based on the points you have.
4. Click calculate to see the distance and the formula used.

The formula explained

• \(d\) = distance between the two points
• \(x_1\) = first point x-coordinate
• \(y_1\) = first point y-coordinate
• \(z_1\) = first point z-coordinate in 3D
• \(x_2\) = second point x-coordinate
• \(y_2\) = second point y-coordinate
• \(z_2\) = second point z-coordinate in 3D

Step by step method

1. Find the difference in each matching coordinate between the two points.
2. Square each difference so negative values do not cancel out.
3. Add the squared differences together, then take the square root of the sum.
4. Read the result as the straight-line distance between the points.

Worked example

Suppose you want to find the straight-line distance between the points (2, 3) and (8, 15) in 2D.

1. Use the 2D formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
2. Substitute the values: \(d = \sqrt{(8 - 2)^2 + (15 - 3)^2}\).
3. Compute the differences and squares: \(d = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180}\).

Tips and common mistakes

• Make sure you match x with x, y with y, and z with z.
• Use the 3D version only when both points have a z-coordinate.
• Do not forget to square each coordinate difference before adding them.
• The answer is a straight-line distance, not a path distance along axes.

Frequently asked questions