What is a compass bearing?

A compass bearing is the horizontal angle measured clockwise from True North (0°) to the direction of travel. It ranges from 0° to 359°. Due North is 0°, due East is 90°, due South is 180° and due West is 270°.

What is the difference between forward and back bearing?

The forward bearing is the compass heading at the origin pointing toward the destination. The back bearing (also called the reciprocal or reverse bearing) is the heading from the destination back to the origin -- approximately 180° different, though on a curved Earth surface the difference is not exactly 180° for long routes.

Why does this calculator use the Haversine formula?

The Haversine formula computes the shortest path between two points on a sphere (a great-circle route). It handles the curvature of the Earth, antipodal points and the International Date Line correctly. The formula is: a = sin²(Δφ/2) + cos φ1 · cos φ2 · sin²(Δλ/2); d = 2R · arcsin(√a), where φ is latitude, λ is longitude and R is Earth radius (6 371 km).

How accurate is the Haversine distance?

The Haversine formula assumes a perfect sphere with radius 6 371 km. Earth is actually a slightly flattened oblate spheroid, so the maximum error is about 0.3% compared to the more precise Vincenty formula. For navigation, hiking, aviation planning and most engineering purposes the Haversine result is more than adequate.

What are decimal degrees and how do I convert from DMS?

Decimal degrees (DD) express latitude/longitude as a single number, e.g., 51.5074°. Degrees-minutes-seconds (DMS) uses 51° 30' 26.64" N. To convert: DD = degrees + (minutes / 60) + (seconds / 3600). The calculator shows the DMS equivalent live as you type.

What is a great-circle route?

A great circle is the largest circle that can be drawn on a sphere, passing through two points and the centre of the sphere. The great-circle route is the shortest path between two points on the surface of the Earth. Aircraft and ships follow great-circle routes to minimise fuel and time.

Bearing Calculator — Gera Tools

A bearing calculator tells you the compass heading and the shortest over-the-surface distance between any two geographic coordinates on Earth. Whether you are planning a hiking route, filing a flight plan, programming a drone waypoint, solving a surveying problem or satisfying curiosity about how far two cities are and in which direction — this tool gives you the answer in under a second, right in your browser.

How it works

The calculator uses two classic geodesy formulas applied to the WGS 84 spherical approximation of Earth (radius 6 371 km).

Haversine distance

The Haversine formula gives the great-circle distance between two points defined by latitude φ and longitude λ:

a  = sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)
c  = 2 · arcsin(√a)
d  = R · c          (R = 6 371 km)

where Δφ = φ₂ − φ₁ and Δλ = λ₂ − λ₁, all in radians. The result is the shortest path along the curved surface — the great-circle distance — expressed in kilometres, miles and nautical miles.

Forward (initial) bearing

The initial bearing is the compass heading at the start point that points along the great-circle path toward the destination:

y = sin(Δλ) · cos φ₂
x = cos φ₁ · sin φ₂  −  sin φ₁ · cos φ₂ · cos(Δλ)
θ = atan2(y, x)          (radians)
bearing = (θ in degrees + 360) mod 360

The result is normalised to the range 0°–359° and then mapped to one of the 16 compass points (N, NNE, NE, ENE … each spanning 22.5°).

Back (final) bearing

The back bearing is the forward bearing computed in the reverse direction (B → A), then increased by 180° and normalised. This represents the heading at which you arrive at the destination — equivalently, the direction back to the origin.

Midpoint

The geographic midpoint on the great circle is found by:

Bx  = cos φ₂ · cos(Δλ)
By  = cos φ₂ · sin(Δλ)
φₘ  = atan2(sin φ₁ + sin φ₂,  √((cos φ₁ + Bx)² + By²))
λₘ  = λ₁ + atan2(By,  cos φ₁ + Bx)

Worked example: London to Paris

Point A (London): 51.5074° N, 0.1278° W → (51.5074, −0.1278)
Point B (Paris): 48.8566° N, 2.3522° E → (48.8566, 2.3522)

Plugging in:

Δφ = −2.6508° = −0.04627 rad
Δλ = 2.48° = 0.04328 rad
a ≈ 0.001090
c ≈ 0.06613 rad
Distance ≈ 341 km (212 mi / 184 NM)
Forward bearing ≈ 156° (SSE) — you head roughly south-southeast from London

This matches commercial flight data: the London-Heathrow to Paris-CDG route tracks between 150° and 160° depending on precise departure/arrival points.

Formula note

The Haversine formula was published by Roger Sinnott in Sky and Telescope (1984) as a numerically stable formulation specifically to avoid cancellation errors for small angular separations. It is accurate to well within 1% across all distances. For very high-precision geodesy (sub-metre accuracy over thousands of kilometres), the Vincenty iterative formula on the WGS 84 ellipsoid is preferred, but for navigation, hiking, drones and mapping applications the Haversine result is entirely sufficient.

Bearings are True bearings (measured from True North), not Magnetic bearings. To convert to Magnetic, subtract the local magnetic declination for your location and date (available from NOAA’s World Magnetic Model).