This azimuth calculator is used to find the horizontal angle between a reference direction and a line from an observer to a point of interest.
θ = atan2(sin(Δλ) * cos(φ2), cos(φ1) * sin(φ2) – sin(φ1) * cos(φ2) * cos(Δλ))
Imagine you’re standing at the center of a compass rose. If you’re facing north and turn 90° to your right (east), the azimuth would be 90°. Similarly, south would be 180°, and west would be 270°.
Azimuth Calculator
| Starting Point | End Point | Latitude 1 | Longitude 1 | Latitude 2 | Longitude 2 | Azimuth |
|---|---|---|---|---|---|---|
| New York | London | 40.7128°N | 74.0060°W | 51.5074°N | 0.1278°W | 51.3° |
| Tokyo | Sydney | 35.6762°N | 139.6503°E | -33.8688°S | 151.2093°E | 188.7° |
| Cairo | Cape Town | 30.0444°N | 31.2357°E | -33.9249°S | 18.4241°E | 181.1° |
| Anchorage | Miami | 61.2181°N | 149.9003°W | 25.7617°N | 80.1918°W | 120.2° |
| Rio de Janeiro | Moscow | -22.9068°S | 43.1729°W | 55.7558°N | 37.6173°E | 48.6° |
| Los Angeles | New Delhi | 34.0522°N | 118.2437°W | 28.6139°N | 77.2090°E | 87.5° |
| Berlin | Beijing | 52.5200°N | 13.4050°E | 39.9042°N | 116.4074°E | 74.1° |
| Buenos Aires | Lima | -34.6037°S | 58.3816°W | -12.0464°S | 77.0428°W | 306.2° |
| Nairobi | Johannesburg | -1.2864°S | 36.8172°E | -26.2041°S | 28.0473°E | 198.5° |
| Madrid | Lisbon | 40.4168°N | 3.7038°W | 38.7223°N | 9.1393°W | 248.9° |
| Moscow | Tokyo | 55.7558°N | 37.6173°E | 35.6762°N | 139.6503°E | 152.5° |
| Dubai | Cairo | 25.276987°N | 55.296249°E | 30.0444°N | 31.2357°E | 333.1° |
| Seattle | Vancouver | 47.6062°N | 122.3321°W | 49.2827°N | 123.1207°W | 337.5° |
| Hanoi | Bangkok | 21.0285°N | 105.8542°E | 13.7563°N | 100.5018°E | 192.1° |
| San Francisco | Mexico City | 37.7749°N | 122.4194°W | 19.4326°N | 99.1332°W | 152.9° |
Azimuth Calculation Formula
The formula for calculating azimuth depends on the given information and the coordinate system used. One common method uses latitude and longitude coordinates:
Azimuth = atan2(sin(Δλ) * cos(φ2),
cos(φ1) * sin(φ2) - sin(φ1) * cos(φ2) * cos(Δλ))Where:
- φ1, λ1 = Latitude and longitude of the starting point
- φ2, λ2 = Latitude and longitude of the end point
- Δλ = Difference in longitude (λ2 – λ1)
This formula yields results in radians, which can be converted to degrees by multiplying by (180/π).
Example: Starting point: 40°N, 75°W
End point: 42°N, 70°W
- Convert to radians
- Apply the formula
- Convert result to degrees
The resulting azimuth would be approximately 58.3°, indicating a northeasterly direction.
How to Calculate Azimuth
- Determine coordinates: Identify the latitude and longitude of both the starting point and the destination.
- Convert to radians: If using degrees, convert all values to radians.
- Apply the formula: Use the azimuth calculation formula mentioned earlier.
- Interpret the result: Convert the output to degrees and adjust if necessary to ensure it falls within the 0-360° range.
Example:
Let’s calculate the azimuth from New York City (40.7128°N, 74.0060°W) to London (51.5074°N, 0.1278°W).
Convert to radians:
- φ1 = 40.7128 × (π/180)
- λ1 = -74.0060 × (π/180)
- φ2 = 51.5074 × (π/180)
- λ2 = -0.1278 × (π/180)
Calculate Δλ = Δλ = λ2 – λ1
Apply the formula
The azimuth from New York to London is approximately 51.3°, indicating a northeastern direction.
What is the Azimuth 0 to 180?
- 0°: Due North
- 45°: Northeast
- 90°: Due East
- 135°: Southeast
- 180°: Due South
The azimuth range from 0° to 180° covers half of the compass rose, representing directions from due north through east to due south.
If a ship captain receives a bearing of 67° to a lighthouse, they know the lighthouse is in the ENE (East-Northeast) direction, falling within the 0-180° azimuth range.
How to Convert Degrees to Azimuth
Converting degrees to azimuth often involves adjusting the angle to fit the 0-360° azimuth scale.
- N45°E = 45° azimuth
- S30°W = 210° azimuth (180° + 30°)
- N80°W = 280° azimuth (360° – 80°)
To convert from a quadrant system:
- Identify the quadrant (NE, SE, SW, NW)
- Use the appropriate formula:
- NE: Azimuth = Angle
- SE: Azimuth = 180° – Angle
- SW: Azimuth = 180° + Angle
- NW: Azimuth = 360° – Angle
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