Distance between two points calculator

Definition and domain

Let A = (x₁, y₁) and B = (x₂, y₂) be points in the Euclidean plane. The distance between two points is the length of the segment joining A to B. In Cartesian coordinates the formula is d = √((x₂ − x₁)² + (y₂ − y₁)²). The differences Δx = x₂−x₁ and Δy = y₂−y₁ are the legs; d is the hypotenuse. The result is a non-negative scalar, not a point.

Domain of this calculator: all finite real coordinates in the fields x₁, y₁, x₂, y₂ (identifiers a–d). Codomain: d ≥ 0 after round(d×1e6)/1e6. When A = B, you get d = 0 — a valid result. An empty field enters as 0; when all four fields are empty, the panel stays in the waiting state. The result unit is the same unit in which you enter the coordinates.

Euclid: length of a segment

In Euclid’s Elements, segment length is a geometric quantity independent of a coordinate frame. Two points determine exactly one segment; its length is non-negative and zero only when the points coincide. Distance satisfies the metric axioms: d(A,B) = d(B,A), d(A,B) = 0 ⇔ A = B, and the triangle inequality d(A,C) ≤ d(A,B) + d(B,C).

This page computes only the Euclidean metric in the plane. It does not introduce Manhattan distance (|Δx|+|Δy|) or Chebyshev distance. Britannica (“Euclidean geometry”) and MathWorld (“Distance”) describe distance as segment length — the same quantity this calculator writes algebraically. In school and engineering use on a map or in CAD, that length is the straight-line path between two points, without detours around obstacles.

Pythagoras: legs Δx and Δy

If you build a rectangle with sides |Δx| and |Δy| from vertex A, the diagonal to B has length √(Δx²+Δy²). That is the Pythagorean theorem: the sum of the squares of the legs equals the square of the hypotenuse. Squared differences drop the sign, so the direction from A to B does not change d: (0,0) and (−3,−4) give the same 5 as (0,0) and (3,4).

The classic 3-4-5 triple: √(9+16) = 5. Doubling gives 6-8-10 and d = 10. Pythagorean relations were known earlier (including the Plimpton 322 context); Euclid gives an axiomatic proof via areas of squares on the sides (I.47). This calculator does not require integer coordinates — fractional values work as well.

Descartes: the coordinate formula

René Descartes (1596–1650) wrote geometry in coordinates: each point is a pair of numbers. Hence the bridge between Euclidean geometry and algebra: d = √((x₂−x₁)²+(y₂−y₁)²). In vector form, d = ‖B − A‖ — the Euclidean norm of the difference of position vectors. With A = (0,0) the formula reduces to the vector length ‖B‖ = √(x₂²+y₂²).

The calculator may use hypot(Δx, Δy), which computes the same square root with better overflow resistance than √(Δx²+Δy²) for very large coordinates. For typical school examples the result is identical. The panel also shows Δx, Δy, and the sum of squares under the root so the Pythagorean step is visible next to d. The calculator works in R²; it does not add a z coordinate.

Full derivation and examples

Step 1. Compute Δx = x₂−x₁ and Δy = y₂−y₁.

Step 2. Form the sum of squares s = Δx² + Δy² (always ≥ 0).

Step 3. Take the non-negative root d = √s, then round(d×1e6)/1e6.

Step 4. Worked examples on this page: (0,0)&(3,4) → 5; (0,0)&(1,1) → 1.414214 (√2 after rounding); (0,0)&(6,8) → 10; (2,1)&(5,5) → 5 (again Δx=3, Δy=4); (−2,3)&(4,−1) → 7.211103 (√52 after rounding). The result display uses a decimal point in US English formatting.

Numerical notes, neighboring models, and limits

The 1e6 round is a presentation contract: a clean 5 stays 5, not 5.000000; √2 appears as 1.414214. Comma and period are equivalent on input. The SVG diagram draws segment AB; the numeric length comes from the formula, not from pixels.

The midpoint returns the point M = ((x₁+x₂)/2, (y₁+y₂)/2), not the scalar d. Distance from a point to a line uses |Ax₀+By₀+C|/√(A²+B²). Vector length from the origin is the same square root with A = (0,0). A two-leg Pythagorean page enters lengths a and b, not four coordinates. This calculator computes only the Euclidean distance between two points. The result is educational.

Sources

Mathematical model: Euclidean distance between two points. Context: Euclid, Elements (segment length); Pythagorean theorem (I.47 / Heath); René Descartes — coordinate form. Sources MathWorld — Distance; Wikipedia — Euclidean distance; Wikipedia — Pythagorean theorem; Britannica — Euclidean geometry; Brannan, Esplen, Gray — Geometry (Cambridge). The calculator does not replace a textbook or a teacher’s check.