Great Circle Calculator

Enter the latitude and longitude of two points to find the great-circle distance, initial and final bearing, and the midpoint along the shortest path over Earth's surface.

Quick Facts

Haversine formula
a = sin²(Δφ/2) + cos φ1 cos φ2 sin²(Δλ/2)
Distance = R · 2·atan2(√a, √(1−a)), using latitude/longitude in radians.
Earth's mean radius
R ≈ 6,371 km (3,959 mi, 3,440 nmi)
The calculator models Earth as a sphere of this average radius.
Great circle vs. rhumb line
Great circle = shortest path
A rhumb line holds a constant bearing but is usually a longer route.

Your Results

Calculated
Great-Circle Distance
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Shortest path over the sphere
Initial Bearing
-
Heading at the start point
Final Bearing
-
Heading on arrival at point 2
Midpoint
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Coordinates halfway along the arc

Ready

Enter two coordinates and press Calculate.

Formula and Method for Great-Circle Distance

A great circle is the largest possible circle that can be drawn on a sphere — its plane always passes through the sphere's center. The arc of a great circle between any two points is the shortest possible path between them along the surface, which is why it is the route ships and aircraft actually follow for long-distance travel. This calculator finds that distance, plus the compass bearings and midpoint, from two latitude/longitude coordinates using the haversine formula.

How the calculation works

Convert both points' latitude (φ) and longitude (λ) to radians. Compute the differences Δφ = φ2 − φ1 and Δλ = λ2 − λ1, then apply the haversine formula: a = sin²(Δφ/2) + cos φ1 · cos φ2 · sin²(Δλ/2), followed by c = 2 · atan2(√a, √(1−a)). Multiplying c by Earth's mean radius R (about 6,371 km) gives the great-circle distance d = R · c. The initial bearing (compass heading leaving point 1) is θ = atan2(sin Δλ · cos φ2, cos φ1 · sin φ2 − sin φ1 · cos φ2 · cos Δλ), normalized to 0–360°. The final bearing is found by running the same formula from point 2 to point 1 and reversing the direction by 180°. The midpoint is the point exactly halfway along the great-circle arc, found with a separate vector-based formula.

Common mistakes

  • Degrees vs. radians: latitude and longitude are entered in degrees, but trigonometric functions need radians. Forgetting to convert (multiply by π/180) gives wildly wrong results.
  • Sign conventions: longitude is negative west of the Prime Meridian and positive east; latitude is negative south of the equator and positive north. Entering the wrong sign can place a point on the opposite side of the planet.
  • Great circle vs. straight line on a map: a great-circle route often looks curved on a flat (Mercator) map even though it is the shortest real-world path — that curvature is a map projection artifact, not a longer route.

Real-world applications

  • Airlines and shipping routes use great-circle paths to minimize fuel and travel time on long-haul journeys.
  • Celestial navigation and orienteering use bearings computed the same way to plot a course between waypoints.
  • GIS, mapping, and geofencing software use the haversine formula to compute distances between GPS coordinates.
  • Radio and satellite communications use great-circle bearings to aim antennas at the shortest path to a target location.

Frequently Asked Questions

What is a great-circle distance?
A great-circle distance is the shortest path between two points on the surface of a sphere, measured along the arc of the "great circle" that passes through both points — a circle whose plane cuts through the sphere's center. On Earth, this is shorter than a straight line drawn on a flat map (a rhumb line) for any two points that are not on the same meridian or the equator.
What formula does this calculator use?
It uses the haversine formula: a = sin²(Δφ/2) + cos φ1 · cos φ2 · sin²(Δλ/2), c = 2 · atan2(√a, √(1−a)), and distance = R · c, where φ1, φ2 are the latitudes, Δφ and Δλ are the differences in latitude and longitude in radians, and R is Earth's mean radius (about 6,371 km).
What is the difference between a great circle and a rhumb line?
A great circle is the shortest path between two points on a sphere and generally appears curved on a flat map. A rhumb line (loxodrome) crosses every meridian at the same angle, so it looks straight on a Mercator map, but it is usually longer than the great-circle route, especially over long distances.
How accurate is the haversine formula?
The haversine formula treats Earth as a perfect sphere, so it is accurate to within about 0.5% of the true distance, which uses an ellipsoidal model (like Vincenty's formulae) for higher precision. For most navigation, aviation, and mapping purposes, the haversine result is more than accurate enough.

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