Spherical Geometry Formulas
Great circle distance and spherical excess formulas for navigation and geodesy.
Calculate distances on a sphere with worked examples.
Great Circle Distance (Haversine Formula)
The great circle distance is the shortest path between two points on the surface of a sphere. On Earth, this is the route that airplanes follow on long-distance flights.
The formula uses latitude and longitude coordinates to compute the angular separation between two points, then multiplies by the sphere's radius to get the actual distance.
For better numerical precision with small distances, the Haversine formula is preferred:
d = 2R × arcsin(√a)
Spherical Excess
In flat (Euclidean) geometry, the angles of a triangle always sum to exactly 180° (π radians). On a sphere, the angles of a triangle always sum to more than 180°. The amount by which the sum exceeds 180° is called the spherical excess.
The area of a spherical triangle is directly related to its spherical excess:
Variables
| Symbol | Meaning |
|---|---|
| d | Great circle distance between two points (meters or km) |
| R | Radius of the sphere (for Earth, R ≈ 6,371 km) |
| φ₁, φ₂ | Latitudes of the two points (in radians) |
| Δλ | Difference in longitudes (in radians) |
| Δφ | Difference in latitudes (in radians) |
| A, B, C | Interior angles of a spherical triangle (in radians) |
| E | Spherical excess (in radians) |
Example 1: Great Circle Distance
Find the great circle distance from New York (40.7128° N, 74.0060° W) to London (51.5074° N, 0.1278° W).
Convert to radians: φ₁ = 40.7128° = 0.7106 rad, φ₂ = 51.5074° = 0.8989 rad
Δλ = |−74.0060 − (−0.1278)|° = 73.8782° = 1.2892 rad
cos(d/R) = sin(0.7106) sin(0.8989) + cos(0.7106) cos(0.8989) cos(1.2892)
= 0.6521 × 0.7826 + 0.7581 × 0.6225 × 0.2756
= 0.5103 + 0.1300 = 0.6403
d/R = arccos(0.6403) = 0.8762 rad
d = 6,371 × 0.8762
d ≈ 5,582 km (about 3,468 miles — the actual flight distance is close to this)
Example 2: Spherical Excess
A spherical triangle on Earth's surface has interior angles of 95°, 80°, and 70°. What is the spherical excess, and what is the area of the triangle?
Sum of angles = 95° + 80° + 70° = 245°
Spherical excess E = 245° − 180° = 65° = 65 × π/180 = 1.1345 radians
Area = R² × E = (6,371)² × 1.1345
Area = 40,589,641 × 1.1345
Area ≈ 46,049,000 km² (about 9% of Earth's surface — a very large triangle)
When to Use These
Spherical geometry is essential for navigation and geoscience.
- Calculating flight distances and routes between cities
- Maritime navigation and plotting ship courses
- Satellite coverage and communication link distances
- Surveying large areas where Earth's curvature matters
- Astronomy and calculating angular separations between stars