How to Find the Distance Between Two Points in 2D and 3D

 

The distance between two points is the length of the straight segment joining them. On a Cartesian coordinate plane, you can calculate that length from the horizontal and vertical coordinate differences. In three dimensions, a third difference accounts for depth or height. The method works with positive, negative and zero coordinates.

The Distance Calculator Between Two Points supports both 2D and 3D coordinates. It also reports Point 2 minus Point 1 and the midpoint. All axes must use the same unit and scale for the reported distance to represent a physical length.

Start with the coordinate differences

For Point 1 (x1, y1) and Point 2 (x2, y2), subtract the first coordinate from the second on each axis. The horizontal difference is dx = x2 - x1, and the vertical difference is dy = y2 - y1. These differences may be positive, negative or zero.

For points (1, 2) and (4, 6), dx is 3 and dy is 4. Imagine a right triangle with horizontal and vertical legs of those lengths. The direct distance between the points is its hypotenuse, so the Pythagorean theorem supplies the next step.

Apply the 2D distance formula

The formula is distance = sqrt(dx squared + dy squared), or sqrt((x2 - x1)^2 + (y2 - y1)^2). In the example, the result is sqrt(3^2 + 4^2) = sqrt(25) = 5 coordinate units.

In the calculator, set Dimensions to 2D (x, y). Enter x1 1, y1 2, x2 4 and y2 6. Select Calculate distance. You should see distance 5, coordinate differences (3, 4) and midpoint (2.5, 4).

Add the third coordinate for 3D

In three dimensions, calculate dz = z2 - z1 and add its square. The formula becomes distance = sqrt(dx^2 + dy^2 + dz^2). This measures the straight segment through Cartesian space, rather than a path that follows the coordinate axes.

Select 3D (x, y, z), which clears the previous coordinates. Enter x1 = 0, y1 = 0, z1 = 0, x2 = 2, y2 = 3 and z2 = 6. The squared differences total 4 + 9 + 36 = 49, giving a distance of 7 coordinate units.

Understand negative coordinates and point order

Negative coordinates work with the same formula. From (-2, -1) to (1, 3), the differences are 3 and 4, so the distance is again 5. You must subtract the full signed value: subtracting -2 is equivalent to adding 2.

Swapping the two points reverses the signs of the differences but does not change their squares or the distance. Identical points have distance zero. Distance is always nonnegative; the coordinate differences preserve direction from the first point toward the second.

Find the midpoint

The midpoint averages the coordinates separately. In 2D, it is ((x1 + x2) / 2, (y1 + y2) / 2). In 3D, include (z1 + z2) / 2 as the third coordinate. For (1, 2) and (4, 6), the midpoint is (2.5, 4), halfway along the connecting segment.

Check units and the meaning of distance

If both axes measure meters using the same scale, the distance is in meters. Mixing feet on one axis with meters on another produces an invalid physical comparison unless you convert first. An image or chart with differently scaled axes also needs adjustment before applying the formula.

Latitude and longitude are angles on the Earth, not ordinary Cartesian distances. Do not enter geographic coordinates here expecting kilometers or miles. Driving distance, routes around obstacles and distances across curved surfaces require other methods.

Allow for numerical rounding

The calculator accepts coordinates from -1000000000 to 1000000000 with up to six decimal places. Results display at most six decimal places; very small nonzero magnitudes appear as a threshold. Calculations retain unrounded intermediate values, but browser arithmetic is approximate. Displayed precision does not make an approximate measurement exact.