Euclidean distance

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 = (p₁, p₂) and q = (q₁, q₂) 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

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

Related pages

• Norm (mathematics)
• Polar coordinates