Coordinate Distance Calculator

Find the distance between two points in 2D or 3D space, plus the midpoint, with step-by-step solutions.

Enter Coordinates

Point 1 (x1, y1)

Point 2 (x2, y2)

Result

Distance
5
units
Point 1 (1, 2)
Point 2 (4, 6)
Distance 5
Midpoint (2.5, 4)
Slope 1.333333

Step-by-Step Solution

d = sqrt((4-1)^2 + (6-2)^2) = 5

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem and calculates the straight-line distance (Euclidean distance) between two points in a coordinate system. It is one of the most fundamental formulas in coordinate geometry and analytic geometry.

In two dimensions, given points P1(x1, y1) and P2(x2, y2), the distance is computed as the square root of the sum of the squared differences in each coordinate.

Distance and Midpoint Formulas

2D Distance

Distance between two points in a plane.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

3D Distance

Distance extended to three-dimensional space.

d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

2D Midpoint

The point exactly halfway between two points.

M = ((x1+x2)/2, (y1+y2)/2)

3D Midpoint

Midpoint in three-dimensional space.

M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)

Slope (2D)

The rate of change between two points in a plane.

m = (y2 - y1) / (x2 - x1)

Manhattan Distance

The sum of absolute differences (city-block distance).

d = |x2-x1| + |y2-y1|

Deriving the Distance Formula

The distance formula comes directly from the Pythagorean theorem. Imagine a right triangle formed by the horizontal and vertical differences between two points. The horizontal leg has length |x2 - x1|, and the vertical leg has length |y2 - y1|. The hypotenuse of this triangle is the straight-line distance between the two points.

By the Pythagorean theorem: c^2 = a^2 + b^2, where c is the distance, a is the horizontal difference, and b is the vertical difference. Taking the square root of both sides yields the distance formula.

Example: Distance Between (1, 2) and (4, 6)

  1. Calculate differences: dx = 4 - 1 = 3, dy = 6 - 2 = 4
  2. Square the differences: dx^2 = 9, dy^2 = 16
  3. Sum the squares: 9 + 16 = 25
  4. Take the square root: sqrt(25) = 5

Applications of the Distance Formula

  • Navigation: Calculating straight-line distances between locations on a map.
  • Physics: Finding displacement vectors and magnitudes.
  • Computer graphics: Collision detection, proximity checks, and rendering calculations.
  • Data science: k-Nearest Neighbors, clustering algorithms, and similarity metrics.
  • Engineering: Structural analysis, surveying, and spatial planning.

Tips for Using the Distance Formula

  • The order of the points does not matter since the differences are squared.
  • Distance is always non-negative; it equals zero only when the two points coincide.
  • For geographic distances on a sphere, use the Haversine formula instead.
  • The midpoint is useful for finding center points and bisectors.