What is a great-circle distance?

A great circle is the largest possible circle that can be drawn on a sphere — its plane passes through the centre of the sphere. The arc of a great circle connecting two surface points is the shortest possible route between them, which is why transoceanic flights appear curved on flat maps but are actually the straight-line path on the globe.

What is the Haversine formula?

The Haversine formula computes the central angle c between two points using a = sin²(Δφ/2) + cos(φ₁)·cos(φ₂)·sin²(Δλ/2) and then c = 2·asin(√a). Multiplying c by the mean Earth radius gives the arc length. It is numerically stable for small distances (unlike the spherical law of cosines) and accurate to within roughly 0.5% because it treats the Earth as a perfect sphere.

What is the Vincenty formula and when should I use it?

Vincenty (1975) iteratively solves the geodesic on the WGS-84 oblate spheroid — the same reference ellipsoid used by GPS. It converges to sub-0.5 mm accuracy for virtually any pair of points. Use it when you need survey-grade precision, when comparing to GPS measurements, or when working near the poles where the Earth deviates most from a sphere. For casual navigation or flight planning, Haversine is perfectly adequate.

What are forward and reverse bearings?

The forward bearing is the initial compass heading you would set off on from Point A to travel the great-circle route to Point B, measured clockwise from north (0°–360°). Because the route curves on a globe, by the time you arrive at Point B the bearing has changed — the reverse bearing is the heading from B back to A, which is not simply 180° plus the forward bearing except along the equator.

How accurate is this calculator?

Haversine mode is accurate to within about 0.5% (roughly 5 km on a 1,000 km route) because it uses a sphere with the IUGG mean radius of 6,371.009 km. Vincenty mode achieves better than 0.5 mm accuracy by modelling the WGS-84 ellipsoid. Neither accounts for terrain elevation, so actual travel distances over mountains or valleys will be greater.

Can I use degrees, minutes, and seconds instead of decimal degrees?

This tool accepts decimal degrees only. To convert from DMS, use the formula decimal = degrees + minutes/60 + seconds/3600. For example, 51° 30′ 26.65″ N converts to 51 + 30/60 + 26.65/3600 = 51.5074°.

Great-Circle Distance Calculator

The great-circle distance is the shortest possible path between any two points on the surface of a sphere, following the curvature of the Earth rather than cutting through it. This calculator implements both the Haversine formula (fast, spherical model, within 0.5%) and the Vincenty inverse formula (WGS-84 ellipsoid, sub-millimetre accuracy), so you can pick the level of precision the task requires.

In addition to the raw distance in kilometres, miles, and nautical miles, the tool computes the forward and reverse initial bearings (shown on interactive compass roses), the geographic midpoint of the route, and a side-by-side comparison of the two formulas so you can see exactly how much the Earth’s oblateness matters for any given pair of coordinates.

How the formulas work

Haversine (spherical Earth)

The Haversine formula derives the central angle c subtended by the two surface points at the Earth’s centre:

a = sin²(Δφ/2) + cos(φ₁) · cos(φ₂) · sin²(Δλ/2)
c = 2 · asin(√a)
d = R · c

where φ is latitude, λ is longitude (both in radians), and R = 6,371.009 km (IUGG mean radius). The name “Haversine” refers to the half-versine trig identity that makes the formula numerically stable even for very short distances, unlike the simpler spherical law of cosines which suffers from floating-point cancellation at small angles.

Vincenty (WGS-84 ellipsoid)

The WGS-84 reference ellipsoid has a semi-major axis of 6,378,137 m, a semi-minor axis of 6,356,752.314 m, and a flattening of 1/298.257. Vincenty’s inverse method iterates on the longitude difference λ until it converges (typically within 4–5 iterations for most routes, at most ~200 for near-antipodal points). The resulting distance can differ from Haversine by up to ~22 km on very long routes and far more near the poles — the comparison table in the tool shows the exact difference for your inputs.

Initial bearing

The initial bearing (also called the forward azimuth) is computed from:

y = sin(Δλ) · cos(φ₂)
x = cos(φ₁) · sin(φ₂) − sin(φ₁) · cos(φ₂) · cos(Δλ)
θ = atan2(y, x)  converted to 0°–360°

Because great-circle routes curve relative to lines of constant latitude (rhumb lines), the bearing changes continuously along the route. A flight from London to New York starts heading west-southwest but arrives heading almost due west — the initial bearing shown is the heading to set at the very start of the journey.

Worked example

London (51.5074° N, 0.1278° W) to New York (40.7128° N, 74.0060° W):

Haversine: 5,570.2 km (3,460.9 mi, 3,007.6 NM)
Vincenty: 5,570.5 km (difference of ~300 m)
Forward bearing: 288.3° (WNW)
Return bearing: 51.2° (NE)
Midpoint: approximately 53.5° N, 40.2° W — over the North Atlantic

The great-circle path passes far north of the straight line you might draw on a Mercator map, swinging up over Scotland and Iceland before curving south toward New York. This northward arc is shorter because the Earth’s circumference is smaller near the poles.

Route	Distance (km)	Forward bearing
London → New York	5,570	288° WNW
Sydney → Tokyo	7,823	60° ENE
Paris → Dubai	5,240	113° ESE
Yerevan → Tbilisi	247	17° NNE

Why this matters

Understanding great-circle distances is essential in aviation (route planning, fuel calculations), maritime navigation (voyage planning, logbook entries), telecommunications (satellite footprint coverage), and any geospatial analysis that involves comparing or sorting locations by distance. Mapping libraries like PostGIS, Turf.js, and GeoPandas all use variants of these same formulas under the hood.