Great-Circle Distance Calculator — Shortest Path on a Sphere
Find the shortest path along Earth's surface between two geographic coordinates using the haversine formula. Enter latitude and longitude in decimal degrees (negative = South/West) and the calculator returns the distance in kilometres, miles and nautical miles, plus the initial compass bearing from Point 1 to Point 2. London to New York is pre-loaded as a worked example.
°
°
°
°
Shortest path along the surface of a spherical Earth (mean radius 6,371 km)
- 1
Δlat in radians
(40.7128 − 51.5074) × π ÷ 180 = -0.188401 - 2
Δlon in radians
(-74.006 − -0.1278) × π ÷ 180 = -1.289418 - 3
Haversine a
sin²(Δlat÷2) + cos(φ₁)·cos(φ₂)·sin²(Δlon÷2) = 0.179236Intermediate haversine value; always in [0, 1]. - 4
Central angle c
2 × atan2(√0.179236, √0.820764) = 0.874309 - 5
Great-circle distance
6,371 × 0.874309 = 5,570.2
How does this calculator work?
The great-circle distance uses the haversine formula: d = 2R · atan2(√a, √(1−a)) where a = sin²(Δlat/2) + cos(lat₁)·cos(lat₂)·sin²(Δlon/2) and R = 6,371 km. London–New York is about 5,570 km. Result is the shortest surface path, not the straight-line distance through Earth's interior.
Formula
How this is calculated
The great-circle distance is the shortest path between two points on the surface of a sphere — it follows the arc of a circle whose plane passes through the centre of the sphere. For Earth, this is the route an aircraft or ship would take ignoring wind and terrain, sometimes called the orthodromic distance.
The haversine formula is a numerically stable version of the spherical law of cosines, derived from the half-angle identities. It computes the central angle c between the two points, then multiplies by Earth's mean radius (6,371 km per IAU 2015). The result uses the WGS-84 mean radius, which introduces an error of at most about 0.5 % because Earth is an oblate spheroid flattened at the poles; for true geodetic accuracy over long baselines, ellipsoidal models such as Vincenty's formulae are needed.
The initial bearing (forward azimuth) is the compass heading at Point 1 pointing toward Point 2 along the great-circle arc. Because great-circle routes curve on a Mercator-projected map (appearing as arcs, not straight lines), the bearing changes continuously along the path — this calculator gives only the starting direction.
Frequently asked questions
Straight-line ('as the crow flies' through the Earth) ignores the curvature of the planet. Great-circle distance measures along the surface — it's what a plane or ship actually travels. For short distances they differ negligibly; for intercontinental routes the difference is significant.
The great-circle path between two points at similar latitudes often passes closer to the poles than it appears on a flat map. A Mercator map makes polar routes look longer because of how it distorts latitude, but on a globe they are genuinely the shortest route.
The haversine formula using a mean Earth radius of 6,371 km is accurate to within about 0.5 % for any point pair. For sub-kilometre surveying accuracy you'd need an ellipsoidal model (e.g. Vincenty's inverse formula over the WGS-84 ellipsoid).
Also known as
TG we-Calculate Editorial Team. (2026). Great-Circle Distance Calculator — Shortest Path on a Sphere [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/great-circle-calculator
TG we-Calculate Editorial Team. "Great-Circle Distance Calculator — Shortest Path on a Sphere." TG we-Calculate. 2026. https://we-calculate.com/calculator/great-circle-calculator.
TG we-Calculate Editorial Team, "Great-Circle Distance Calculator — Shortest Path on a Sphere," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/great-circle-calculator
@misc{wecalculate_great_circle_calculator, title = {Great-Circle Distance Calculator — Shortest Path on a Sphere}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/great-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?