$Distance Between Points$

Distance Between Points

Find the distance between two points from their coordinates in 2D and 3D with an exact answer and a step-by-step solution.

Space: in the plane (x, y) in space (x, y, z)

Coordinates are whole numbers, decimals or fractions: 3, −1.5, 2/3.

Test yourself

The distance between points calculator finds the distance between two points from their coordinates in the plane and in space. The answer is exact — with a simplified root (for example, √8 = 2√2) — and approximate. It also shows the coordinates of vector AB and of the midpoint.

Distance formula

Case	Formula
On a line	d = \|x_B − x_A\|
In the plane	d = √((x_B − x_A)² + (y_B − y_A)²)
In space	d = √((x_B − x_A)² + (y_B − y_A)² + (z_B − z_A)²)

Example 1. A(−1, 2), B(3, −1).
d = √(4² + (−3)²) = √25 = 5.

Example 2. A(1, 2, 3), B(4, 6, 15).
d = √(3² + 4² + 12²) = √169 = 13.

The formula follows from the Pythagorean theorem: the coordinate differences are the legs of a right triangle, and segment AB is its hypotenuse. The distance equals the length of vector AB and does not depend on the order of the points.