Azimuth Calculator (initial and final bearing between two coordinates)

Understanding the azimuth calculator — concept and principle

The azimuth is the direction from one place to another measured clockwise from true north. This calculator finds the initial great-circle azimuth, the azimuth on arrival, the reverse azimuth, the constant rhumb-line bearing and the distance between two points given in latitude and longitude.

Measured clockwise from true north, Busan lies at 144.09° from Seoul, roughly south-east.

How it is calculated

Initial azimuth θ = atan2(sin Δλ · cos φ₂, cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ), normalised to 0–360°.
Reverse azimuth: the same formula from point 2 back to point 1; final azimuth = reverse − 180°.
Rhumb-line bearing = atan2(Δλ, Δψ), with Δψ = ln[tan(π/4 + φ₂/2) ÷ tan(π/4 + φ₁/2)].
Distance by the haversine formula with a mean Earth radius of 6,371.0 km.
NATO mils = degrees × 6,400 ÷ 360.

The principle

The shortest path on a sphere is a great circle, and its compass heading changes along the way except along the equator or a meridian. The atan2 formula gives the starting direction from spherical trigonometry. A rhumb line instead crosses every meridian at the same angle, so it can be followed with one compass setting, but it is slightly longer. Over short distances the two are almost identical.

Worked example

Default inputs: Seoul (37.5665° N, 126.9780° E) to Busan (35.1796° N, 129.0756° E).

Initial azimuth ≈ 144.09° (south-east), or about 2,562 NATO mils.
Final azimuth on arrival ≈ 145.33°.
Reverse azimuth from Busan ≈ 325.33°.
Rhumb-line bearing ≈ 144.72°.
Great-circle distance ≈ 325.11 km.

Each point is the calculator re-run with only “Point 2 latitude (°)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Point 2 latitude (°)	Initial azimuth from true north (°)
8.79	175.7
17.6	174.16
26.4	170.39
35.2	144.09
44	13.25
52.8	4.82
70.4	1.3

How to read the result

Bearings are true, not magnetic. To follow them with a compass, apply the local magnetic declination. Over a few hundred kilometres the great-circle and rhumb-line bearings differ by less than a degree; over intercontinental distances the difference can be large.

Common mistakes

Swapping latitude and longitude.
Using west longitudes as positive.
Forgetting magnetic declination when using a compass.
Assuming the reverse azimuth is exactly the initial azimuth ± 180°.

Keep in mind: Uses a spherical Earth; differences from ellipsoidal (WGS 84) results are small but matter for surveying. Do not rely on this tool for navigation safety.

Frequently asked questions

What is the difference between azimuth and bearing?

Azimuth is measured 0–360° clockwise from north; bearings are sometimes given in quadrant form such as S 36° E.

Why does the azimuth change along a great circle?

Meridians converge towards the poles, so the angle between the path and each meridian changes.

What is a rhumb line?

A path that keeps a constant compass bearing, crossing all meridians at the same angle.

Sources: Snyder, J. P. (1987). Map Projections—A Working Manual. USGS Professional Paper 1395 · Sinnott, R. W. (1984). Virtues of the Haversine. Sky and Telescope, 68(2), 159

Checked 9 Nov 2026. Spherical-trigonometry formulas for the initial great-circle azimuth, rhumb-line bearing and haversine distance (standard formulas); mean Earth radius 6,371.0088 km (IUGG mean radius, standard value); NATO mil = 1/6400 of a circle.

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