Great Circle Distance
Shortest distance between two points on the Earth for long-range navigation.
Formula
Variables
Example
The shortest air route from New York to Tokyo curves near the Arctic — a great-circle path.
How it works
This formula gives the shortest surface route between two points on a spherical Earth by measuring the central angle between their position vectors and multiplying by the Earth’s radius. The quantity inside the arccosine is the dot product of the two unit vectors written in latitude–longitude form, so it captures how far apart the points are in 3D geometry rather than in flat map coordinates. Because a great circle is the intersection of the Earth with a plane through its center, its arc is the minimum-distance path over the sphere.
When to use it
Use it for long-range navigation, especially in aviation, shipping, and satellite ground-track planning, when straight-line map distances would be misleading. It is most appropriate when Earth can be approximated as a sphere and you need the shortest path along the surface rather than through the Earth.
Worked example, step by step
Find the great-circle distance between Los Angeles (34.0522° N, 118.2437° W) and Honolulu (21.3069° N, 157.8583° W), using R = 6371 km. Step 1: Convert to radians. φ1 = 34.0522° = 0.5943 rad φ2 = 21.3069° = 0.3719 rad Δλ = 157.8583° - 118.2437° = 39.6146° = 0.6914 rad Step 2: Compute the trig terms. sin φ1 = sin(0.5943) ≈ 0.5592 sin φ2 = sin(0.3719) ≈ 0.3633 cos φ1 = cos(0.5943) ≈ 0.8290 cos φ2 = cos(0.3719) ≈ 0.9317 cos Δλ = cos(0.6914) ≈ 0.7708 Step 3: Substitute into the formula. inside arccos = (0.5592)(0.3633) + (0.8290)(0.9317)(0.7708) inside arccos ≈ 0.2031 + 0.5953 = 0.7984 Step 4: Find the central angle. θ = arccos(0.7984) ≈ 0.6450 rad Step 5: Multiply by Earth’s radius. d = 6371 × 0.6450 ≈ 4109 km Final answer: The great-circle distance is about 4,110 km.
Common mistakes to avoid
- Using degrees directly in the trigonometric functions instead of converting to radians
- Forgetting that longitude difference should reflect the actual angular separation, including west/east signs or wraparound across 180°
- Treating the result as a straight-line-through-Earth distance rather than a surface arc length
Did You Know?
The shortest route between two cities curves toward the pole — why flights to Asia arc over the Arctic.
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