Bearing and Distance Calculator

Determine a destination point’s geographic coordinates from a starting point, bearing, and distance.

Destination Coordinates

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This result is calculated using Vincenty’s formulae on a spherical earth model, providing the destination coordinates after traveling a set distance along a constant bearing from a starting point.

What is Calculating Distances Using Bearing?

Calculating distances using bearing is a fundamental technique in navigation and geodesy. It involves determining a future geographic position (latitude and longitude) based on a known starting point, a direction of travel (bearing), and the distance to be covered. This calculation, often called a “forward geodetic problem,” is essential for pilots, sailors, surveyors, and anyone needing to plot a course over the Earth’s surface. Unlike simple planar geometry, it must account for the planet’s curvature, making the math more complex.

Common misunderstandings arise from the difference between true bearing, magnetic bearing, and grid bearing. This calculator uses true bearing, where 0° is True North. You must also select the correct units, as using kilometers instead of miles will produce a significantly different result. Our haversine formula explained article provides more background on spherical calculations.

The Bearing and Distance Calculation Formula

To find the destination point, we treat the Earth as a sphere and use spherical trigonometry. Given a starting point (lat₁, lon₁), a bearing (θ), and a distance (d), we can calculate the destination point (lat₂, lon₂).

The formulas are as follows, where R is the Earth’s radius:

lat₂ = asin( sin(lat₁) × cos(d/R) + cos(lat₁) × sin(d/R) × cos(θ) )

lon₂ = lon₁ + atan2( sin(θ) × sin(d/R) × cos(lat₁), cos(d/R) − sin(lat₁) × sin(lat₂) )

Note that all latitude, longitude, and bearing values must be converted to radians for the trigonometric functions to work correctly. The result is then converted back to degrees. For more on this, our introduction to geodesy is a great resource.

Calculation Variables

Variable	Meaning	Unit	Typical Range
lat1, lon1	Starting Latitude & Longitude	Decimal Degrees	-90 to +90 (lat), -180 to +180 (lon)
θ	True Bearing	Degrees	0 to 360
d	Distance	Kilometers or Miles	Any positive number
R	Earth’s Radius	Kilometers or Miles	~6371 km or ~3959 mi
lat2, lon2	Destination Latitude & Longitude	Decimal Degrees	Calculated result

Practical Examples

Example 1: Flight Path

Imagine a short flight taking off from Chicago O’Hare (approx. 41.97° N, 87.90° W) and flying for 250 miles on a bearing of 110°.

• Inputs:
  • Starting Latitude: 41.97
  • Starting Longitude: -87.90
  • Bearing: 110°
  • Distance: 250 miles
• Results:
  • Destination Latitude: 40.75° N
  • Destination Longitude: -83.79° W

This shows that proper bearing and distance calculation is crucial for aviation.

Example 2: Sailing Route

A boat leaves a port in Lisbon, Portugal (approx. 38.72° N, 9.14° W) and sails for 50 kilometers on a bearing of 225° (South-West).

• Inputs:
  • Starting Latitude: 38.72
  • Starting Longitude: -9.14
  • Bearing: 225°
  • Distance: 50 km
• Results:
  • Destination Latitude: 38.40° N
  • Destination Longitude: -9.55° W

This demonstrates how a marine navigation calculator uses the same principles.

How to Use This Bearing and Distance Calculator

1. Enter Starting Coordinates: Input the latitude and longitude of your starting point in decimal degrees. Use positive values for North/East and negative for South/West.
2. Set the Bearing: Provide the true bearing, which is the angle from True North clockwise. A bearing of 0° is North, 90° is East, 180° is South, and 270° is West.
3. Specify the Distance: Enter the total distance you plan to travel.
4. Select Units: Choose whether the distance is in kilometers or miles. This is a critical step for an accurate result. The calculator automatically uses the correct Earth radius for the chosen unit.
5. Review Results: The calculator will instantly display the destination latitude and longitude. The visual chart helps you understand the path taken.

Key Factors That Affect Bearing Calculations

• Earth Model: This calculator uses a spherical model of the Earth. For extremely high-precision surveying over long distances, an ellipsoidal model (like WGS84) is more accurate, as the Earth is slightly flattened at the poles.
• Bearing Type: We use True Bearing. If you have a Magnetic Bearing (from a compass), you must first correct for magnetic declination to find the True Bearing.
• Unit Accuracy: Ensuring your distance unit (km or miles) is correct is paramount. A mistake here will lead to a large error in the destination.
• Path Type: The calculation assumes travel along a rhumb line (loxodrome), which is a line of constant bearing. This is not the shortest distance between two points on a sphere (a great circle), but it is easier to navigate. Our calculator for understanding great circles shows the difference.
• Input Precision: The more decimal places you provide for your starting coordinates, the more precise the final calculated point will be.
• Altitude: Our calculations are at sea level. For high-altitude applications (like aerospace), the Earth’s radius would need to be adjusted by the altitude, which would slightly alter the distance calculation.

Frequently Asked Questions (FAQ)

1. What is the difference between bearing and azimuth?

In many contexts, bearing and azimuth are used interchangeably. Both refer to a horizontal angle measured clockwise from a north base line. This tool uses the term bearing as is common in general navigation.

2. Why are there different results for kilometers and miles?

The calculation depends on the Earth’s radius (R). When you select ‘Kilometers’, we use R ≈ 6371 km. When you select ‘Miles’, we use R ≈ 3959 miles. Using the wrong unit means the scale of your calculation is incorrect.

3. Does this calculator account for magnetic declination?

No. This is a pure geodetic calculator that requires a True Bearing. You must convert any magnetic bearing to true bearing before using the tool.

4. How accurate is this calculation?

The accuracy is very high for most navigational purposes. It is based on a spherical Earth model. For global-scale surveying where millimeters matter, more complex ellipsoidal models are required.

5. Can I use this for finding a location by bearing from two points?

No, this is a “forward” calculator (point + bearing + distance = new point). To find your location from bearings to two known points (a resection), you would need a different tool or triangulation method.

6. What does “NaN” mean if it appears in the result?

NaN stands for “Not a Number”. It will appear if you enter non-numeric text into the input fields or leave them blank. Please ensure all inputs are valid numbers.

7. How can I find the coordinates of a starting point?

You can use online mapping services or a GPS device. Many tools, including our coordinate finder, can help you find the latitude and longitude of any location.

8. What happens at the poles or the 180° meridian?

The formulas used handle these edge cases correctly. The math allows longitude to wrap around from +180° to -180°, and calculations near the poles remain consistent.

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