# Euclidean distance

conventional distance in mathematics and physics

In Euclidean geometry, the Euclidean distance is the usual distance between two points p and q. This distance is measured as a line segment. The Pythagorean theorem can be used to calculate this distance.[1][2]

## Euclidean distance on the plane

Euclidean distance in R2

In the Euclidean plane, if p = (p1p2) and q = (q1q2) then the distance is given by[3]

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

This is equivalent to the Pythagorean theorem, where legs are differences between respective coordinates of the points, and hypotenuse is the distance.

Alternatively, if the polar coordinates of the point p are (r1, θ1) and those of q are (r2, θ2), then the distance between the points is

${\displaystyle d(\mathbf {p} ,\mathbf {q} )={\sqrt {r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta _{1}-\theta _{2})}}.}$

## References

1. Weisstein, Eric W. "Distance". mathworld.wolfram.com. Retrieved 2020-09-01.
2. "Distance Between 2 Points". www.mathsisfun.com. Retrieved 2020-09-01.
3. "Distance Between 2 Points". www.mathsisfun.com. Retrieved 2020-09-01.