What is the Haversine formula?

The Haversine formula calculates the shortest distance between two points on a sphere (the great-circle distance) from their latitudes and longitudes. It is named after the haversine trigonometric function hav(theta) = sin²(theta/2). The formula is a = sin²(dLat/2) + cos(lat1)·cos(lat2)·sin²(dLon/2); c = 2·asin(sqrt(a)); d = R·c, where R is the Earth radius (6 371 km).

How accurate is the Haversine distance?

It is accurate to within roughly 0.3-0.5% for any two surface points. The error arises because the Earth is an oblate spheroid, not a perfect sphere; the polar radius (6 357 km) is about 21 km shorter than the equatorial radius (6 378 km). For aviation, shipping, and most GIS applications the Haversine result is entirely sufficient. For geodetic surveying requiring millimetre precision, use Vincenty's formulae or the Karney method.

What is decimal-degree format?

Decimal degrees express latitude and longitude as a single signed number rather than degrees-minutes-seconds. Latitude ranges from -90 (South Pole) to +90 (North Pole); longitude from -180 to +180 (negative = West). To convert from DMS, use decimal = degrees + (minutes / 60) + (seconds / 3600). For example, 51° 30' 26" N becomes 51.5074, and 0° 7' 40" W becomes -0.1278.

What is the difference between great-circle distance and straight-line distance?

A straight-line (Euclidean) distance would pass through the Earth's interior. Great-circle distance is the shortest path along the surface of the sphere, which is the arc of a circle whose centre is the Earth's centre. For points more than a few hundred kilometres apart the difference becomes significant. Aeroplanes and ships follow great-circle routes (subject to winds and currents) because they are the shortest surface paths.

What do the forward and reverse bearings mean?

The forward bearing is the initial compass direction you must travel from Point A to reach Point B (measured clockwise from true north). The reverse bearing is the initial direction from Point B back to Point A. Because Earth is curved, these two bearings are not simply 180° apart except along the equator or a meridian. For example, flying from London to New York you head roughly west-southwest (283°), but the return bearing from New York is roughly east-northeast (51°).

Why does the calculator use 6 371.009 km as the Earth radius?

6 371.009 km is the International Union of Geodesy and Geophysics (IUGG) mean radius, defined as the arithmetic mean of the equatorial and two polar radii. It is the standard reference radius for single-sphere approximations and is used by most navigation software, Google Maps distance calculations, and the OpenLayers mapping library.

Haversine Distance Calculator

The Haversine Distance Calculator finds the great-circle distance — the shortest path across the surface of the Earth — between any two geographic coordinates. Enter two pairs of decimal-degree latitude/longitude values and the tool returns the distance in kilometres, miles, nautical miles, and metres, plus the initial compass bearing in both directions. Everything runs inside your browser; no data is uploaded anywhere.

How it works

The Earth is treated as a sphere with the IUGG mean radius R = 6 371.009 km. The great-circle distance between two points (lat1, lon1) and (lat2, lon2) — all in radians — is found with the Haversine formula:

a = sin²(dLat/2) + cos(lat1) · cos(lat2) · sin²(dLon/2)

c = 2 · asin(sqrt(a))

d = R · c

where dLat = lat2 − lat1 and dLon = lon2 − lon1.

The intermediate value a is the square of half the chord length between the two points; it is always in [0, 1], which prevents asin from returning NaN due to floating-point rounding. The central angle c (in radians) is then multiplied by the Earth radius to get the arc length.

Initial bearing from A to B uses the four-quadrant arctangent of the cross-track components:

y = sin(dLon) · cos(lat2)

x = cos(lat1)·sin(lat2) − sin(lat1)·cos(lat2)·cos(dLon)

bearing = atan2(y, x) (converted to 0–360°)

The reverse bearing (B → A) is calculated identically with the coordinates swapped, not just by adding 180° — because on a curved surface the two headings differ except along meridians and the equator.

Worked example: London to New York

Property	Value
Point A	51.5074° N, 0.1278° W
Point B	40.7128° N, 74.0060° W
dLat	−10.7946° = −0.18840 rad
dLon	−73.8782° = −1.28942 rad
a	0.17924
c	0.87431 rad
Distance	5 570.2 km (3 461.2 mi, 3 007.7 NM)
Forward bearing	288.3° (WNW)
Reverse bearing	51.2° (NE)

The Haversine result agrees with published great-circle distances for this route to within 1 km. The asymmetry of the bearings (283° outbound vs 51° return) illustrates how initial headings diverge on a sphere.

Coordinate entry tips

Decimal degrees are the simplest format: 51.5074 for 51° 30’ 26” N.
West longitudes and south latitudes are negative: -74.006 for 74° W.
Most smartphone GPS apps and Google Maps display decimal degrees in the “What’s here?” popup — tap the coordinates to copy them.
The tool includes four quick-preset routes (London → New York, Sydney → Tokyo, Paris → Dubai, Yerevan → Tbilisi) to demonstrate typical results at a glance.

Limitations and alternatives

The Haversine formula assumes a spherical Earth. The real Earth is an oblate spheroid, so the maximum error is about 0.5% for antipodal points. For most navigation, mapping, and logistics use-cases this is negligible. If you need sub-metre geodetic precision — such as survey-grade GPS processing — use Vincenty’s inverse formula or the Karney (2013) geodesic method, which model the WGS-84 ellipsoid.

Unit conversions used: 1 km = 0.621 371 mi (exactly); 1 nautical mile = 1.852 km (exactly, since 1954); 1 km = 1 000 m.