# Euclidean Distance

In mathematics, the **Euclidean distance** between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance.

![Euclidean distance between two points](https://upload.wikimedia.org/wikipedia/commons/5/55/Euclidean_distance_2d.svg)

## Distance formulas

### One dimension

The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates

![One dimension formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d75418dbec9482dbcb70f9063ad66e9cf7b5db9)

### Two dimensions

![Two dimensions formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0157084fd89f5f3d462efeedc47d3d7aa0b773)

### Higher dimensions

In three dimensions, for points given by their Cartesian coordinates, the distance is

![Three dimensions formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d13a40a7b203b455ae6d4be8b3cce898bda625)

Example: the distance between the two points `(8,2,6)` and `(3,5,7)`:

![3-dimension example](https://www.mathsisfun.com/algebra/images/dist-2-points-3d.svg)

In general, for points given by Cartesian coordinates in `n`-dimensional Euclidean space, the distance is

![n-dimensional formula](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ef4fe055b2a51b4cca43a05e5d1cd93f758dcc)

## References

- [Euclidean Distance on MathIsFun](https://www.mathsisfun.com/algebra/distance-2-points.html)
- [Euclidean Distance on Wikipedia](https://en.wikipedia.org/wiki/Euclidean_distance)