What is the azimuth?
By the US Army definition, the term azimuth describes the angle created by two lines: one joining your current position and the North Pole, and the one joining your current position and the distant location. Azimuth is always measured clockwise!
For example, a point lying east from you would have an azimuth of 90°, but a point lying west from you — 270°.
The azimuth is used when indicating a position in the sky too: it marks the horizontal direction. The altitude indicates the vertical direction, varying from 0° (the horizon) to 90° (the zenith). The point opposite to the zenith is called the nadir.
This azimuth calculator lets you calculate the azimuth from the latitude and longitude of two points. It tells you which direction you need to point your compass to and what the shortest distance between the two points of known geographical coordinates is. This is not a spherical coordinates calculator — it deals with geographical coordinates only!
Azimuth formula
To give the location of a point relative to your current position, you need two values: the azimuth and the distance. Because Earth is (approximately) a sphere, the distance is the shortest travel distance between the two points — the "as-the-crow-flies" great-circle distance.
The distance d is found with the Haversine formula:
a = sin²(Δφ/2) + cos(φ₁) · cos(φ₂) · sin²(Δλ/2) d = 2R · atan2(√a, √(1−a))
The azimuth θ uses the same latitudes and longitudes:
θ = atan2( sin(Δλ)·cos(φ₂),
cos(φ₁)·sin(φ₂) − sin(φ₁)·cos(φ₂)·cos(Δλ) )
Where:
- φ₁ — latitude of the initial point (positive for N, negative for S)
- φ₂ — latitude of the final point (positive for N, negative for S)
- λ₁ — longitude of the initial point (positive for E, negative for W)
- λ₂ — longitude of the final point (positive for E, negative for W)
- Δφ = φ₂ − φ₁ and Δλ = λ₂ − λ₁
- R — radius of the Earth (R = 6,371 km ≈ 3,958.8 mi)
The result of the atan2 function is converted to degrees and normalized to the
0°–360° range. Input latitudes and longitudes in decimal degrees notation.
To convert degrees-minutes-seconds to decimal degrees, use a degrees-minutes-seconds converter first.
How to calculate the azimuth: an example
Let's find the azimuth and distance from New York to London.
- New York (Empire State Building): 40.7484°, −73.9857°
- London (Big Ben): 51.5007°, −0.1246°
- Convert all coordinates to radians.
- Compute Δλ = λ₂ − λ₁ and plug the values into the azimuth formula above.
- The forward azimuth is ≈ 51.4° (NE) — point your compass roughly north-east.
- The great-circle distance is ≈ 5,570 km (3,461 mi).
The calculator also reports the reverse azimuth — the bearing you would follow travelling back from Point 2 to Point 1 — and the 16-point compass abbreviation (N, NNE, NE, …).
Metric vs. US units
- Metric: the distance is shown in kilometers (km) and meters (m).
- US / Imperial: the distance is shown in miles (mi), feet (ft), and yards (yd).
- Nautical miles (nmi): always shown as well — used in aviation and maritime navigation; 1 nautical mile = 1,852 m = 1.15078 statute miles.
The azimuth itself is an angle, so it is always expressed in degrees regardless of the unit system.
FAQs
Is azimuth measured clockwise or counter-clockwise?
Azimuth is always measured clockwise from north (0°). So east is 90°, south is 180°, and west is 270°.
What is the difference between azimuth and bearing?
In this context they are the same: the initial bearing (or forward azimuth) is the compass direction from Point 1 towards Point 2, measured clockwise from north in the 0°–360° range.
Why does the reverse azimuth differ from azimuth ± 180°?
On a great circle the direction gradually changes as you travel, so the exact reverse azimuth (Point 2 → Point 1) is recalculated rather than simply flipped by 180°. Over short distances the two are almost identical, but they can differ noticeably over long, curved routes.
How accurate is the distance?
The Haversine formula treats Earth as a perfect sphere and is accurate to within about 0.3%. For high-precision geodetic work, the Vincenty formula or the WGS-84 ellipsoid model is used instead.
Can I enter coordinates with N/S/E/W notation?
This calculator uses decimal degrees. Apply the sign to convert: S and W are negative. Example: 40°44′N → +40.7333°; 74°00′W → −74.0000°.