How to Calculate the Distance Between Two Coordinates (Haversine Explained)

You have two GPS coordinates — how far apart are they? The answer is a single formula used by pilots, sailors, and every mapping app: the haversine formula, which computes the great-circle distance across the Earth's surface. Here is how it works, with a real example you can verify.

Straight Line on a Sphere: the Great Circle

On a flat map, the shortest path between two points is a straight line. On a sphere, it is an arc of a great circle — the circle you get by slicing the sphere through its center and both points. This is why long-haul flights between, say, Paris and Tokyo arc over Siberia instead of following a straight line on the map: the "curved" route is actually the shortest one.

The Haversine Formula

Given two points with latitude φ and longitude λ (in radians):

a = sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)
d = 2R · asin(√a)

where R is the Earth's mean radius, 6,371,008.8 meters. The formula treats the Earth as a sphere, which introduces an error of only about 0.5% versus the true ellipsoidal shape — negligible for almost every practical purpose.

Worked Example: Seoul to Busan

• Seoul: 37.5665, 126.9780
• Busan: 35.1796, 129.0756

Step by step:

1. Δφ = 35.1796 − 37.5665 = −2.3869° → −0.04166 rad
2. Δλ = 129.0756 − 126.9780 = 2.0976° → 0.03661 rad
3. a = sin²(−0.02083) + cos(0.6557) · cos(0.6140) · sin²(0.01831) ≈ 0.000651
4. d = 2 × 6,371,008.8 × asin(√0.000651) ≈ 325.1 km

The actual driving distance between the two cities is roughly 400 km — the straight-line figure is always the floor, never the road estimate.

Initial Bearing: Which Way to Go

The same two points also define an initial bearing — the compass direction you would face at the start:

θ = atan2( sin Δλ · cos φ₂ , cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ )

For Seoul → Busan this gives about 144°, i.e. southeast. Note the word initial: on long routes the bearing changes continuously as you follow the great circle. A flight from Paris to Los Angeles starts out heading northwest even though Los Angeles is southwest of Paris on a flat map.

Why Real Routes Are Longer

The great-circle distance is the "as the crow flies" number. Real journeys add:

• Roads and rails that follow terrain, valleys, and property lines
• Shipping lanes constrained by coastlines, canals, and traffic separation
• Flight paths shaped by air corridors, weather, and jet streams

As a rule of thumb, driving distance in developed road networks runs about 20–40% above the great-circle distance.

Skip the Math

If you just need the number, our free distance calculator does all of this in one click: click two points on the map (or type coordinates or addresses), and get the great-circle distance in kilometers, miles, or nautical miles plus the initial bearing. To convert coordinate formats first, use the coordinate converter.