Distance Formula Calculator
Our free coordinate geometry calculator solves distance formula problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
The Euclidean distance formula is derived from the Pythagorean theorem. The differences along each axis form the legs of a right triangle, and the distance is the hypotenuse. Manhattan distance sums absolute differences. Chebyshev distance takes the maximum absolute difference.
Worked Examples
Example 1: Distance Between Two 2D Points
Problem:Find the distance between A(1, 2) and B(4, 6).
Solution:dx = 4 - 1 = 3\ndy = 6 - 2 = 4\nEuclidean = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5\nManhattan = |3| + |4| = 7\nChebyshev = max(|3|, |4|) = 4\nMidpoint = ((1+4)/2, (2+6)/2) = (2.5, 4)
Result:Euclidean: 5 | Manhattan: 7 | Chebyshev: 4 | Midpoint: (2.5, 4)
Example 2: 3D Distance Calculation
Problem:Find the distance between P(1, 2, 3) and Q(4, 6, 8).
Solution:dx = 4 - 1 = 3\ndy = 6 - 2 = 4\ndz = 8 - 3 = 5\nEuclidean = sqrt(9 + 16 + 25) = sqrt(50) = 7.0711\nManhattan = 3 + 4 + 5 = 12\nChebyshev = max(3, 4, 5) = 5\nMidpoint = (2.5, 4, 5.5)
Result:Euclidean: 7.0711 | Manhattan: 12 | Chebyshev: 5 | Midpoint: (2.5, 4, 5.5)
Frequently Asked Questions
What is the distance formula and how is it derived?
The distance formula calculates the straight-line (Euclidean) distance between two points in a coordinate space. For 2D points (x1, y1) and (x2, y2), the formula is d = sqrt((x2-x1)^2 + (y2-y1)^2). It is derived directly from the Pythagorean theorem by treating the horizontal difference (x2-x1) and vertical difference (y2-y1) as the two legs of a right triangle, with the distance as the hypotenuse. In 3D, the formula extends to d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2) by applying the Pythagorean theorem twice. This formula is one of the most fundamental tools in analytic geometry and is used extensively in physics, engineering, and computer science.
What is the difference between Euclidean, Manhattan, and Chebyshev distance?
These three distance metrics measure separation between points in different ways. Euclidean distance is the straight-line distance (the hypotenuse), representing the shortest path between two points. Manhattan distance (also called taxicab or L1 distance) sums the absolute differences along each axis, representing the distance traveled along a grid like city blocks: d = |x2-x1| + |y2-y1|. Chebyshev distance (also called chessboard distance) takes the maximum absolute difference along any axis: d = max(|x2-x1|, |y2-y1|), representing the number of king moves in chess. Each metric defines different shaped unit circles: a circle for Euclidean, a diamond for Manhattan, and a square for Chebyshev.
How does the distance formula extend to three dimensions?
The 3D distance formula adds a z-component to the standard 2D formula: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). This can be derived by applying the Pythagorean theorem in two steps. First, find the distance in the xy-plane: d_xy = sqrt((x2-x1)^2 + (y2-y1)^2). Then treat d_xy and the z-difference as legs of another right triangle: d = sqrt(d_xy^2 + (z2-z1)^2). The formula generalizes to any number of dimensions: for n-dimensional points, d = sqrt(sum of (xi2-xi1)^2 for all i). This generalization is called the Euclidean norm and is fundamental to machine learning, where data points often exist in high-dimensional feature spaces.
What is the squared distance and when should I use it?
The squared distance is simply the distance formula without the square root: d^2 = (x2-x1)^2 + (y2-y1)^2. While it does not represent the actual geometric distance, it preserves the ordering of distances (if d1 > d2, then d1^2 > d2^2 for non-negative distances). This makes squared distance useful in optimization and comparison problems where you only need to know which distance is larger, not the actual values. Computing squared distance is faster because it avoids the relatively expensive square root operation. In machine learning, algorithms like k-nearest neighbors often use squared distance for efficiency. Least-squares regression minimizes the sum of squared distances from data points to the fitted line.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy