Great Circle Distance
Calculate the shortest path between two points on Earth's surface.
Used in aviation and maritime route planning.
The Formula
d = R × arccos(sin(φ₁) × sin(φ₂) + cos(φ₁) × cos(φ₂) × cos(Δλ))
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