Distance Formula Calculator (Distance & Midpoint)

The distance formula is the Pythagorean theorem applied to coordinates: the horizontal and vertical gaps between two points are the legs of a right triangle, and the distance is its hypotenuse. This calculator works in 2D or 3D, gives the answer in simplified radical form (such as 3√5) as well as a decimal, and also finds the midpoint, with each substitution written out.

How to use this tool

1. Enter the x and y coordinates of both points.
2. For 3D, also enter both z values; leave them empty for a flat plane.
3. Read the distance, its exact radical form and the midpoint.
4. Follow the steps line to see each subtraction and square.

Formula

d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²) · midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)

Leave out the z terms in 2D. The radical is simplified by taking square factors out of the square root: √45 = √(9 × 5) = 3√5. Decimal coordinates are scaled to whole numbers first, so (0.1, 0.1) gives √2/10 exactly.

Example

From (1, 2) to (7, 5), the gaps are 6 and 3, so d = √(36 + 9) = √45 = 3√5 ≈ 6.7082. The midpoint is (4, 3.5). In 3D, from (1, 2, 3) to (4, 6, 15) the gaps 3, 4 and 12 give d = √169 = 13.