Great Circle Distance
Calculate the shortest path between two points on Earth using the Haversine formula.
Used in aviation, shipping, and GPS for accurate great circle routing.
The Formula
The great circle is the shortest path between two points on a sphere. Airplanes follow great circle routes because they save significant distance on long flights.
Variables
| Symbol | Meaning |
|---|---|
| d | Great circle distance (km or miles) |
| R | Radius of Earth (6,371 km or 3,959 miles) |
| φ₁, φ₂ | Latitude of points 1 and 2 (radians) |
| Δλ | Absolute difference in longitude (radians) |
Example 1
Distance from Paris (48.86°N, 2.35°E) to New York (40.71°N, 74.01°W)
φ₁ = 48.86° = 0.8528 rad, φ₂ = 40.71° = 0.7106 rad
Δλ = 76.36° = 1.3330 rad
cos(d/R) = sin(0.8528)sin(0.7106) + cos(0.8528)cos(0.7106)cos(1.3330)
d ≈ 5,837 km (3,626 miles)
Example 2
Distance from the North Pole (90°N) to the Equator at any longitude
Δφ = 90°, so the arc length = R × π/2
d = 6,371 × 1.5708
d ≈ 10,008 km (one quarter of Earth's circumference)
When to Use It
Use the great circle distance when:
- Planning the most efficient flight or shipping route
- Comparing the straight-line and actual travel distances
- Understanding why polar routes are shorter for some flights
- Calculating distances on any sphere or planet
Key Notes
- The spherical law of cosines formula loses numerical precision for very short distances (under ~1 km) because arccos of values close to 1 magnifies floating-point errors — for short distances, use the Haversine formula, which is numerically stable at all scales
- Great circle routes often cross polar regions — flights from Los Angeles to London arc over Greenland because that is the shortest path on a sphere, even though it looks "wrong" on a Mercator flat map where great circles appear as curves
- Earth is not a perfect sphere but an oblate spheroid flattened at the poles — the sphere model introduces errors up to ~0.5%; the WGS-84 ellipsoid model (used by GPS) combined with the Vincenty formula gives accuracy to 0.5 mm
- All angles must be in radians for this formula — convert degrees to radians by multiplying by π/180; using degrees directly without conversion produces a completely wrong answer without any obvious error signal