Euclidean Distance Calculator

Calculate the straight-line distance between two points in 2D, 3D, or n-dimensional space with component-wise breakdown.

Select Dimensions & Enter Coordinates

Point P1 (x1, y1) and Point P2 (x2, y2)

Result

Euclidean Distance
5
units
Point P1(1, 2)
Point P2(4, 6)
Dimensions2
Squared Distance25

Step-by-Step Solution

d = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5

Understanding Euclidean Distance

Euclidean distance is the "ordinary" straight-line distance between two points in Euclidean space. Named after the ancient Greek mathematician Euclid, it is the most intuitive and commonly used distance metric. In two dimensions, it corresponds to the length of the line segment connecting two points, and can be derived from the Pythagorean theorem.

The general formula for Euclidean distance in n-dimensional space between points P = (p1, p2, ..., pn) and Q = (q1, q2, ..., qn) is: d(P, Q) = sqrt(SUM(pi - qi)^2 for i = 1 to n).

Distance Formulas by Dimension

1D Distance

Distance on a number line is simply the absolute difference.

d = |x2 - x1|

2D Distance

The classic distance formula from the Pythagorean theorem.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

3D Distance

Extended to three-dimensional space for spatial problems.

d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

n-D Distance

Generalized to any number of dimensions for data science and machine learning.

d = sqrt(SUM(xi - yi)^2)

Midpoint Formula

The point exactly halfway between two endpoints.

M = ((x1+x2)/2, (y1+y2)/2)

Manhattan Distance

Alternative metric: sum of absolute differences (taxicab distance).

d = |x2-x1| + |y2-y1|

Applications of Euclidean Distance

Euclidean distance is fundamental in numerous fields. In machine learning, it is used in k-nearest neighbors (KNN), k-means clustering, and similarity measures. In physics, it calculates displacement. In computer graphics, it determines pixel distances and collision detection. In navigation, it estimates straight-line distance between locations.

Tips for Distance Calculations

  • Euclidean distance is always non-negative and equals zero only when both points are identical.
  • It satisfies the triangle inequality: d(A,C) is always less than or equal to d(A,B) + d(B,C).
  • For high-dimensional data, consider that distances tend to become less meaningful (curse of dimensionality).
  • When comparing distances, you can compare squared distances to avoid the square root computation.
  • Ensure all coordinates use the same units before computing distance.