How to use this calculator
- Enter or choose start latitude.
- Enter or choose start longitude.
- Enter or choose destination latitude.
- Enter or choose destination longitude.
- The initial bearing updates as you type. Use Copy result or Share to keep it.
Method
θ = atan2(sin Δλ · cos φ₂, cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ), normalised to 0–360°; final bearing = (bearing from 2 to 1 + 180°) mod 360°
- φ₁, φ₂
- start and destination latitudes
- Δλ
- difference in longitude
- θ
- bearing clockwise from true north
Worked example
- Δλ = 90°, so sin Δλ = 1 and cos Δλ = 0.
- θ = atan2(cos 35°, cos 35° × sin 35°) = atan(1 ÷ sin 35°) = atan(1.743447) ≈ 60.1624°.
- Both points share a latitude, so by symmetry the final bearing is 180° − 60.1624° = 119.8376°.
From 35°N 45°E to 35°N 135°E you set off on 60.16° (ENE) and arrive heading 119.84° (ESE).
What the result means
A great-circle route changes heading continuously. A constant compass heading (a rhumb line) is easier to steer but longer, except along the equator or a meridian.
Assumptions and limits
- Spherical Earth (mean radius 6,371.0088 km for the distance).
- Bearings are from true north, not magnetic north; add or subtract the local magnetic declination for a compass.
- Undefined at a pole and for identical or antipodal points.
- Ellipsoidal (WGS-84) azimuths can differ by a fraction of a degree.
Questions
Why is the final bearing different from the initial one?
Meridians converge towards the poles, so a great circle crosses each one at a different angle. Only routes along the equator or a meridian keep a constant bearing.
Is the return bearing just the initial bearing plus 180°?
Only on a flat map. On the sphere the bearing for the return trip is the final bearing plus 180°, which generally differs.