About

Azimuth is the angular measurement between a reference direction (true north) and a line connecting two points on Earth's surface, measured clockwise from 0° to 360°. Miscalculating azimuth by even 1° over a 100km traverse introduces lateral error exceeding 1.7km. This tool computes the initial (forward) bearing θ using the spherical law on the WGS-84 ellipsoid approximation (R = 6371km), and derives the back azimuth and great-circle distance via the Haversine formula. It accepts both decimal degrees and degrees-minutes-seconds notation. Note: results assume a spherical Earth model. For sub-meter geodetic work over long baselines, use Vincenty's formulae on the reference ellipsoid instead.

Formulas

The forward azimuth (initial bearing) from point A to point B on a sphere is derived from spherical trigonometry:

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

The result is normalized to the range [0°, 360°) by: θ_norm = (θ ⋅ 180π + 360) mod 360.

The back azimuth is computed as: θ_back = (θ_norm + 180) mod 360.

Great-circle distance uses the Haversine formula:

a = sin²(Δφ2) + cos(φ₁) ⋅ cos(φ₂) ⋅ sin²(Δλ2)

d = 2 ⋅ R ⋅ atan2(√a, √1 − a)

Where φ₁, φ₂ = latitudes of points A and B in radians. λ₁, λ₂ = longitudes in radians. Δφ = φ₂ − φ₁. Δλ = λ₂ − λ₁. R = 6371km (mean Earth radius). θ = forward azimuth in degrees.

Reference Data

Cardinal / Intercardinal	Abbreviation	Azimuth (°)	Azimuth (rad)	Compass Rose Sector
North	N	0 / 360	0 / 2π	N
North-Northeast	NNE	22.5	0.3927	NNE
Northeast	NE	45	0.7854	NE
East-Northeast	ENE	67.5	1.1781	ENE
East	E	90	π2	E
East-Southeast	ESE	112.5	1.9635	ESE
Southeast	SE	135	2.3562	SE
South-Southeast	SSE	157.5	2.7489	SSE
South	S	180	π	S
South-Southwest	SSW	202.5	3.5343	SSW
Southwest	SW	225	3.9270	SW
West-Southwest	WSW	247.5	4.3197	WSW
West	W	270	3π2	W
West-Northwest	WNW	292.5	5.1051	WNW
Northwest	NW	315	5.4978	NW
North-Northwest	NNW	337.5	5.8905	NNW
Magnetic Declination Reference (approximate, epoch 2025)
New York, USA	-	−13.0° (West)
London, UK	-	−0.5° (West)
Tokyo, Japan	-	−7.6° (West)
Sydney, Australia	-	+12.6° (East)
Moscow, Russia	-	+11.4° (East)
São Paulo, Brazil	-	−21.8° (West)
Cairo, Egypt	-	+4.0° (East)
Mumbai, India	-	−1.1° (West)

Frequently Asked Questions

Forward azimuth (initial bearing) is the angle measured at point A looking toward point B. Back azimuth is the bearing measured at point B looking back toward point A. On a sphere, the back azimuth is NOT simply the forward azimuth plus 180° from A's perspective along the entire path - it equals (forward + 180) mod 360 only at the destination point. For short distances the difference is negligible, but over baselines exceeding 1000 km, the two values diverge measurably due to meridian convergence.

This calculator outputs true azimuth, referenced to geographic (true) north. A magnetic compass reads magnetic north, which differs by the local magnetic declination. To convert true azimuth to a magnetic bearing, subtract the declination if it is east (positive), or add its absolute value if it is west (negative). For example, in New York (declination approximately −13°), a true azimuth of 90° corresponds to a magnetic bearing of about 103°. Declination varies by location and changes over time at roughly 0.1° per year.

The spherical model (radius R = 6371 km) simplifies computation while yielding azimuth accuracy within approximately 0.3% for most practical distances. The WGS-84 ellipsoid has a flattening of about 1/298.257, causing deviations primarily in north-south bearings over long baselines. For sub-meter survey-grade work, Vincenty's inverse solution or Karney's algorithm on the ellipsoid is required. This tool is appropriate for navigation, orienteering, radio antenna alignment, and planning where 0.1° precision suffices.

Yes. Toggle the input format to DMS mode, and the calculator accepts degrees (0-90 for latitude, 0-180 for longitude), minutes (0-59), and seconds (0-59.999) with a hemisphere selector (N/S for latitude, E/W for longitude). Internally, the values are converted to decimal degrees: DD = degrees + minutes/60 + seconds/3600, with a sign applied for S or W hemispheres.

When point A and point B are identical (or within floating-point tolerance of approximately 1×10⁻⁹ degrees), the azimuth is mathematically undefined because atan2(0, 0) = 0. The calculator returns 0° by convention and flags the distance as 0 km. In practice, GPS receivers exhibit similar behavior, defaulting to north when stationary.

The Haversine formula on a sphere of radius 6371 km produces distance errors up to 0.5% compared to the Vincenty ellipsoidal solution. For a 10,000 km great-circle path, this translates to a maximum error of roughly 50 km. For distances under 100 km, the error is typically under 500 m. The Haversine formula never suffers from convergence failures, unlike Vincenty's method near antipodal points.