Distance Between Cities

Trigonometry · Axiom Academy

LESSON Distance Between Cities Discover how airlines calculate the shortest flight paths using spherical trigonometry and great circles. You're planning a flight from New York to Tokyo. Looking at a flat map, the shortest path seems to go straight east across the Atlantic and Asia. But airlines don't fly that way—why not? But curves north—saving 1,000+ miles! The secret? Earth is a sphere, not flat! The shortest path is a great circle —and trigonometry calculates these distances. Select two cities below to see the great circle distance—the shortest path on Earth's surface. Watch how the path curves! To calculate the great circle distance between two points on a sphere, we use the Haversine formula —a beautiful application of spherical trigonometry. d = distance between the two points R = Earth's radius (≈ 6,371 km) φ₁, φ₂ = latitude of point 1 and point 2 λ₁, λ₂ = longitude of point 1 and point 2 The haversine function converts angular differences into a form suitable for distance calculation. Use inverse haversine to find the angular distance between points. The formula uses sine and cosine to account for Earth's curvature—this is why paths that look straight on a map are actually curved! Now that you understand great circle distances, let's test your knowledge! Which path is shorter from Los Angeles to Dubai? Which great circle path is shorter?

This is the written version of the interactive lesson above. See the full Trigonometry course.