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

[math]\displaystyle{ d(\mathbf{p},\mathbf{q})=\sqrt{(q_1-p_1)^2 + (q_2-p_2)^2}. }[/math]

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

[math]\displaystyle{ d(\mathbf{p},\mathbf{q})=\sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos(\theta_1 - \theta_2)}. }[/math]

Euclidean Distance Media

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

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  Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard

Related pages

• Norm (mathematics)
• Polar coordinates