Last updated: 2026-05-19

The distance between two points

The distance between two points is the length of the straight line segment connecting them. Given points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the Euclidean distance formula is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This is the fundamental measure of "how far apart" two locations are in a flat (2D) coordinate plane. It appears throughout algebra, geometry, physics, computer graphics, and data science.

Derivation from the Pythagorean theorem

The distance formula is a direct application of the Pythagorean theorem. Connecting P₁ to P₂ forms a right triangle:

• The horizontal leg runs from P₁ to the point directly below (or above) P₂. Its length is |x₂ − x₁| = |Δx|.
• The vertical leg runs from that corner up (or down) to P₂. Its length is |y₂ − y₁| = |Δy|.
• The hypotenuse is the segment P₁P₂ — the distance being measured.

The Pythagorean theorem gives d² = Δx² + Δy², so $d = \sqrt{\Delta x^2 + \Delta y^2}$. Squaring removes the absolute values, so the sign of Δx or Δy never affects the result.

Worked Example

Problem: Find the distance between A = (−2, 3) and B = (4, −5).

Step 1 — compute Δx and Δy: $\Delta x = 4 - (-2) = 6 \qquad \Delta y = -5 - 3 = -8$

Step 2 — apply the formula: $d = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

The two points are exactly 10 units apart. This is a scaled version of the 3-4-5 Pythagorean triple (multiply by 2: 6-8-10).

Δx and Δy: Direction vs. Distance

This calculator also reports Δx and Δy separately:

• Δx = x₂ − x₁ is the signed horizontal displacement. A positive value means P₂ is to the right of P₁; a negative value means P₂ is to the left.
• Δy = y₂ − y₁ is the signed vertical displacement. Positive means P₂ is above P₁.

The distance d is always non-negative, but Δx and Δy can be negative depending on the relative positions of the two points.

Order of the points

Distance is symmetric: swapping P₁ and P₂ changes the sign of Δx and Δy but not d, because squaring converts both to positive values. The distance from A to B equals the distance from B to A.

Extension to 3D (and Beyond)

In three dimensions, with points P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂):

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

The reasoning is the same — form a right-angled box, and the space diagonal follows the Pythagorean theorem applied twice. In n dimensions the formula adds one squared difference for each additional coordinate.

Euclidean vs. Taxicab Distance

The Euclidean distance is the straight-line distance — it assumes movement in any direction is possible. The taxicab (Manhattan) distance measures travel along a grid:

$d_{\text{taxicab}} = |\Delta x| + |\Delta y|$

In a city where travel is restricted to city blocks (up/down and left/right), the taxicab distance is the practical minimum travel distance. Euclidean distance is always ≤ taxicab distance, with equality only when the two points share an x- or y-coordinate (the path is a single horizontal or vertical segment).

Common Applications

Coordinate geometry. Finding the length of a line segment, verifying that three points form a right triangle (check whether d(A,B)² + d(B,C)² = d(A,C)²), or computing the radius of a circle given center and perimeter point.

Navigation and mapping. On a local scale, the Euclidean formula approximates "as the crow flies" distances directly from coordinate differences. For global distances, the haversine formula replaces it to account for Earth's curvature.

Computer graphics. Collision detection, picking (clicking on objects), and shader calculations all compute point-to-point distances constantly. The squared distance d² = Δx² + Δy² is often used directly to avoid the expensive square root when only comparisons are needed.

Machine learning. The k-nearest neighbors algorithm classifies a point by finding the k training examples with the smallest Euclidean distances to it. Many clustering methods (k-means, DBSCAN) use Euclidean distance as their default metric.