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.

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

$d(𝐩,𝐪)=\sqrt{(q_1-p_1{)}^2+(q_2-p_2{)}^2}.$

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

$d(𝐩,𝐪)=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos ({\theta }_1-{\theta }_2)}.$

Related pages

• Norm (mathematics)
• Polar coordinates