Great-Circle Distance
Shortest distance between two points on the globe.
Formula
Variables
Example
Used to plot the shortest ocean crossing between two ports.
How it works
This formula gives the shortest path along the Earth's surface between two locations, measured as an arc of a great circle rather than a straight line through space. It comes from spherical trigonometry: the term inside the arccosine is the dot product of the two position vectors on a sphere, which reveals the central angle between them. Multiplying that angle by the Earth's radius converts the angle into a surface distance.
When to use it
Use this when you need an accurate long-distance separation on a globe, especially for navigation, aviation, satellite links, or any problem where Earth’s curvature matters. It is most useful for routes that span large regions or cross oceans, where flat-map approximations become noticeably wrong.
Worked example, step by step
Find the great-circle distance between Lisbon (38.7223° N, 9.1393° W) and New York City (40.7128° N, 74.0060° W), using R = 6371 km. Convert to radians: φ1 = 38.7223° = 0.6752 rad φ2 = 40.7128° = 0.7106 rad Δλ = |−74.0060° − (−9.1393°)| = 64.8667° = 1.1321 rad Compute the spherical cosine term: sinφ1 sinφ2 = sin(0.6752)sin(0.7106) ≈ 0.6240 × 0.6523 = 0.4070 cosφ1 cosφ2 cosΔλ = cos(0.6752)cos(0.7106)cos(1.1321) ≈ 0.7814 × 0.7580 × 0.4256 = 0.2520 Add them: 0.4070 + 0.2520 = 0.6590 Take arccos: arccos(0.6590) ≈ 0.8510 rad Multiply by Earth radius: d = 6371 × 0.8510 ≈ 5423 km Final answer: the great-circle distance is about 5420 km.
Common mistakes to avoid
- Using degrees directly in the trigonometric functions instead of converting to radians first
- Forgetting that Δλ is the difference in longitude, with sign handled consistently before taking the cosine
- Treating the result as a straight-line distance through Earth rather than a surface distance along the sphere
Did You Know?
Great-circle routes look curved on flat maps but are the true shortest path — why flights over oceans arc north.
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