What is the Haversine formula?

The Haversine formula calculates the shortest path (great-circle arc) between two points on a sphere. Given latitudes φ₁ and φ₂, and the longitude difference Δλ, it first computes a = sin²(Δφ/2) + cos(φ₁)·cos(φ₂)·sin²(Δλ/2), then the central angle c = 2·asin(√a), and finally the distance d = R·c where R = 6 371 km is the mean Earth radius. The name comes from the trigonometric identity hav(θ) = sin²(θ/2).

How accurate is this calculator?

The Haversine result is accurate to within about 0.5% for most distances on Earth because it models the Earth as a perfect sphere. For the highest accuracy over long baselines, geodesists use the Vincenty formula or Karney's algorithm on the WGS-84 ellipsoid, which reduce the error to sub-millimetre precision. For aviation, navigation and everyday use Haversine is more than sufficient.

What do positive and negative coordinates mean?

Latitude is positive for north (0° to 90° N) and negative for south (0° to -90° S). Longitude is positive for east (0° to 180° E) and negative for west (0° to -180° W). So London is approximately 51.5° N, -0.13° W, expressed as 51.5074, -0.1278 in decimal degrees.

Can I enter coordinates in degrees-minutes-seconds (DMS)?

Yes. The calculator accepts formats such as "40° 26' 46.302" N" or "40:26:46.302N". For negative (south/west) values include a minus sign or the S/W suffix. Decimal degrees like 40.4462 are also accepted and are the most common format exported by GPS devices and mapping apps.

What is the bearing and why are forward and return bearings different?

The bearing is the compass direction you would travel at the starting point to follow the great-circle arc. The forward bearing is the direction from A to B; the return bearing is the direction from B back to A. They differ because great-circle paths curve across the globe — the path from London to New York heads roughly north-west but arrives heading south-west, so the return bearing is not simply 180° opposite the outgoing one (except along the equator or meridians).

What is the geographic midpoint?

The geographic midpoint is the point equidistant along the great-circle arc between A and B. It is calculated by converting both points to 3-D Cartesian vectors, averaging them, and converting back to latitude/longitude. For routes that do not cross the anti-meridian (180°) this gives an intuitive point on the Earth's surface roughly halfway between the two cities.

Coordinate Distance Calculator

Planning a flight route, writing a script to cluster GPS points, verifying a geofence, or just satisfying curiosity about how far apart two cities really are? This coordinate distance calculator does all of that in one step — enter two latitude/longitude pairs and instantly see the great-circle distance, forward and return bearings, and the geographic midpoint, with a full breakdown of every formula step.

How it works

The tool implements the Haversine formula, the standard algorithm for computing the shortest path between two points on a sphere (the great-circle arc). It uses Earth’s mean radius of 6 371.0088 km (WGS-84 spherical approximation).

Given two points with latitudes φ₁, φ₂ and longitudes λ₁, λ₂ (all in radians):

a = sin²((φ₂−φ₁)/2) + cos(φ₁) · cos(φ₂) · sin²((λ₂−λ₁)/2)

c = 2 · asin(√a)

d = R · c

where d is the distance and R = 6 371.0088 km.

The calculator also derives:

Kilometres, miles and nautical miles — 1 km = 0.621371 mi = 0.539957 nmi.
Initial (forward) bearing — the compass direction at Point A to head towards Point B, using atan2(sin(Δλ)·cos(φ₂), cos(φ₁)·sin(φ₂) − sin(φ₁)·cos(φ₂)·cos(Δλ)), mapped to 0–360°.
Return bearing — the direction from B back to A; not simply 180° opposite because great-circle paths curve.
Geographic midpoint — computed via 3-D Cartesian vector averaging: convert each point to (x, y, z), average, then convert back to latitude/longitude.

Worked example — London to Paris

Field	Value
Point A (London)	51.5074° N, 0.1278° W
Point B (Paris)	48.8566° N, 2.3522° E
Δφ	−2.6508°
Δλ	+2.4800°
a	0.00175…
c	0.05291… rad
Distance	337.3 km / 209.6 mi / 182.2 nmi
Initial bearing	148.3° (SSE)
Return bearing	328.9° (NNW)
Midpoint	~50.18° N, 1.12° E

The forward bearing of ~148° is south-south-east from London, which matches looking at a map. The return bearing of ~329° is north-north-west from Paris back to London.

Why Haversine and not flat-Earth Pythagoras?

For small distances (under ~20 km) the Euclidean approximation is acceptable, but errors compound rapidly as distances grow. London to New York is roughly 5 570 km; treating the Earth as flat gives a result that is off by hundreds of kilometres. The Haversine formula accounts for the curvature of the Earth and stays accurate to within 0.5% for all distances, which is far better than GPS accuracy for most applications.

Formula note

The “Haversine” name comes from the trigonometric identity hav(θ) = sin²(θ/2), which avoids the numerical instability of earlier spherical-law-of-cosines approaches at very short distances. The formula was popularised for navigation in the 19th century and remains the go-to method in GIS, GPS firmware, and spatial databases (PostGIS uses it internally for ST_DistanceSphere).