In 2D geometry, use the Euclidean distance formula: d = √((x₂−x₁)² + (y₂−y₁)²). This is derived directly from the Pythagorean theorem — the distance is the hypotenuse of a right triangle formed by the horizontal (Δx) and vertical (Δy) differences. For a 3D version, add a third term: d = √(Δx²+Δy²+Δz²). For real-world GPS coordinates, the Haversine formula accounts for Earth’s curvature. The calculator above handles all three modes automatically.

The Haversine formula calculates the great-circle distance between two points on a sphere given their latitudes and longitudes. “Haversine” refers to the haversine function: hav(θ) = sin²(θ/2). The formula: a = sin²(Δlat/2) + cos(lat₁)·cos(lat₂)·sin²(Δlon/2); c = 2·atan2(√a, √(1−a)); d = R·c, where R is Earth’s mean radius (6,371 km). It’s accurate to within about 0.3% for most distances, with errors increasing slightly for antipodal points.

The 3D distance formula extends the 2D Pythagorean theorem to three dimensions: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). It calculates the straight-line Euclidean distance through 3D space. Example: distance from point (0,0,0) to (1,2,2) = √(1²+2²+2²) = √(1+4+4) = √9 = 3. This formula is fundamental in 3D computer graphics, physics simulations, robotics, and CAD engineering.

Latitude measures the north-south position from the equator (0°) to the poles (±90°). Positive values are north, negative south. Longitude measures east-west position from the Prime Meridian in Greenwich (0°) to ±180°. Positive values are east, negative west. Together, any location on Earth can be described as (latitude, longitude) — for example, London is approximately (51.5074°N, −0.1278°W), or in decimal degrees: (51.5074, −0.1278).

For short distances (under ~50km), the flat-Earth approximation gives negligible errors — the curvature is too subtle to matter. As distances increase, the great-circle path (following the Earth’s surface) diverges meaningfully from a straight-line calculation. For long distances like London to Sydney (~17,000 km), the Haversine result is accurate to within about 0.3%, while a simple flat calculation would be completely wrong. The Haversine formula used in this calculator assumes a spherical Earth — professional geodesy uses the ellipsoidal WGS84 model (Vincenty formula) for the highest accuracy.