Distance Formula

Using the Pythagorean theorem to compute two-dimensional Euclidean distance

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. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.

The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.

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. Thus if $p$ and $q$ are two points on the real line, then the distance between them is given by:

$$d(p,q)=|p-q|.$$

A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:

$$d(p,q)={\sqrt {(p-q)^{2}}}.$$

In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.

Two dimensions

In the Euclidean plane, let point $p$ have Cartesian coordinates $(p_{1},p_{2})$ and let point $q$ have coordinates $(q_{1},q_{2})$. Then the distance between $p$ and $q$ is given by:

$$d(p,q)={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}}}.$$

This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from $p$ to $q$ as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.

It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of $p$ are $(r,\theta )$ and the polar coordinates of $q$ are $(s,\psi )$, then their distance is given by the law of cosines:

$$d(p,q)={\sqrt {r^{2}+s^{2}-2rs\cos(\theta -\psi )}}.$$

When $p$ and $q$ are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used:

$$d(p,q)=|p-q|.$$

Higher dimensions

Deriving the $n$-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem

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

$$d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}}.$$

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

$$d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}}}.$$

Licensing

Content obtained and/or adapted from:

• Euclidean distance, Wikipedia under a CC BY-SA license