A Beginner's Guide to Great-Circle Distance and GPS Coordinates

Open a flight tracker for a long-haul route — say, New York to Tokyo — and the flight path curves noticeably north, arcing up toward the pole rather than running in a straight horizontal line across the map. That curve isn’t the airline taking a scenic route. It’s the shortest possible path. Understanding why requires remembering something a flat map makes easy to forget: the Earth is a sphere.

Why straight lines on a map aren’t the shortest path

Any flat map necessarily distorts a spherical surface — there’s no way to unroll a sphere onto a plane without stretching or cutting something (this is a real, proven mathematical fact, not a limitation of current mapmaking). The common Mercator-style world map most people picture stretches high-latitude regions dramatically, which is exactly why Greenland looks comparable in size to Africa on that projection despite being about 1/14th its actual area.

Because of this distortion, a straight line drawn on a flat map is not generally the shortest real-world path between two points — that shortest path is called a great-circle route, and it’s defined as the path along the circle you’d get by slicing the sphere through both points and the Earth’s center. For most long-distance routes, the great-circle path curves toward the nearest pole compared to what a flat map’s straight line would suggest.

Latitude and longitude, precisely

Every location on Earth is described by two coordinates: latitude (how far north or south of the equator, from -90° at the South Pole to +90° at the North Pole) and longitude (how far east or west of the Prime Meridian, from -180° to +180°). Together they pin down an exact point.

Coordinates appear in two common formats. Decimal degrees (like 51.5074, -0.1278 for London) is what GPS devices and mapping APIs use internally. Degrees-minutes-seconds (like 51°30'27"N) is the traditional format, where each degree splits into 60 minutes and each minute into 60 seconds — the same base-60 subdivision clocks use, for the same historical reason (both trace back to Babylonian mathematics). The DMS to Decimal Calculator on this site converts between the two directly.

The haversine formula

The haversine formula calculates great-circle distance between two latitude/longitude points on a sphere:

a = sin²(Δφ/2) + cos(φ1) × cos(φ2) × sin²(Δλ/2)
c = 2 × atan2(√a, √(1−a))
distance = R × c

Where φ is latitude, λ is longitude (both in radians), and R is Earth’s mean radius (6,371 km). The formula looks intimidating, but it’s doing something conceptually simple: converting the angular separation between two points on a sphere into an arc length, using trigonometry that correctly accounts for the sphere’s curvature — unlike a naive flat-plane distance formula, which would be wrong by an increasing margin the farther apart the two points are.

As a check: London (51.5074°N, 0.1278°W) to Paris (48.8566°N, 2.3522°E) computes to about 343.6 km — matching the commonly cited straight-line distance between the two cities.

Bearing: which direction to head

Distance alone doesn’t tell you which way to go. Initial bearing is the compass direction, measured clockwise from true north, that points along the start of the great-circle path. For London to Paris, that bearing works out to roughly 148° — southeast, which matches intuition looking at a map. The important caveat: on a long route, following a single constant compass bearing does not trace the great-circle path — it traces a different curve called a rhumb line, which is longer. True great-circle navigation requires the bearing to continuously change along the route, which is exactly why long-haul flight paths curve the way they do.

Midpoint: not just the average

A common mistake is finding the “midpoint” between two coordinates by simply averaging their latitudes and longitudes separately. Because of the sphere’s curvature, this isn’t the true geographic midpoint except in special cases (like two points on the same meridian) — the actual great-circle midpoint requires its own trigonometric formula, which the GPS Midpoint Calculator on this site computes directly, useful for finding a genuinely fair meeting point between two locations.

A limitation worth knowing

The haversine formula treats Earth as a perfect sphere using its mean radius. In reality, Earth is very slightly flattened at the poles (an oblate spheroid) — for most everyday purposes, including trip planning and general distance estimates, this makes a negligible difference. High-precision surveying and geodesy use ellipsoidal models (like WGS84) that account for the flattening, trading simplicity for the extra accuracy that specialized use cases need.

Try it directly

The GPS Coordinate Distance Calculator computes the exact haversine formula above for any two coordinates you enter — a fast way to build real intuition for how map distortion, bearing, and great-circle geometry all connect.