Distance Calculator —
2D & 3D distance formula.

Distance guide

The distance formula: Pythagoras in coordinate geometry.

The distance between two points in a coordinate plane is one of the most fundamental results in all of geometry. It appears in algebra, calculus, physics, computer graphics, data science, and machine learning — wherever you need to quantify how far apart two things are in a structured space. Everything starts with the Pythagorean theorem.

Deriving the 2D distance formula

Given two points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), we can construct a right triangle:

• The horizontal leg has length |Δx| = |x₂ − x₁|
• The vertical leg has length |Δy| = |y₂ − y₁|
• The hypotenuse is the distance d we want

Applying the Pythagorean theorem (a² + b² = c²) with a = |Δx|, b = |Δy|, c = d:

The squares eliminate the need for absolute values, squaring a negative number gives a positive result.

Worked example (2D)

Find the distance between P₁ = (1, 2) and P₂ = (4, 6):

This is a 3-4-5 right triangle. The horizontal and vertical legs are 3 and 4 units respectively, and the hypotenuse — the actual distance — is exactly 5.

Extending to 3D

Adding a third dimension is straightforward: the distance in 3D is the hypotenuse of a right triangle whose legs are the 2D distance in the xy-plane and the vertical separation Δz:

This extends naturally to any number of dimensions n:

In machine learning, this formula computes the Euclidean distance between two data points in high-dimensional feature space, a key operation in k-nearest neighbours, k-means clustering, and SVMs.

Midpoint formula

The midpoint M between P₁ and P₂ is the arithmetic mean of the coordinates:

Geometrically, M lies on the line segment P₁P₂ exactly halfway between the two points. In 3D, add (z₁+z₂)/2 for the z-coordinate of the midpoint.

Slope and the line equation (2D)

Once you know two points, you can determine the equation of the line passing through them using the slope formula:

and then the slope-intercept form with one of the points:

Special cases: when Δx = 0 the line is vertical (x = x₁, slope is undefined) and when Δy = 0 the line is horizontal (y = y₁, slope = 0).

Angle of the line

The angle θ that the line makes with the positive x-axis is:

This uses the two-argument arctangent function, which correctly handles all four quadrants and returns an angle in (−180°, +180°]. The slope is related to the angle by m = tan(θ).

Manhattan distance

Also called taxicab geometry, L₁ distance, orcity-block distance, the Manhattan distance sums the absolute differences rather than using squares:

This is the distance you'd travel between two city blocks if you can only move along streets (no diagonals). The Manhattan distance is always ≥ the Euclidean distance, with equality when the two points share a coordinate (Δx = 0 or Δy = 0). In sparse high-dimensional spaces (e.g., text data), L₁ distance sometimes outperforms L₂.

Unit vector (direction cosines)

The unit vector from P₁ to P₂ gives the direction of travel without encoding the distance:

Its components are called direction cosines in 3D — the cosines of the angles the vector makes with each axis. The length (magnitude) of a unit vector is always 1.

Real-world applications