Bearing calculator

Worked example: Denver to Boston

On the great circle from the Colorado capitol to the Massachusetts capitol you start on 73.0° (ENE) and arrive on 95.7° (E). The path is 1,765 miles long.

Holding one heading of 84.2° instead follows a rhumb line of 1,776 miles, 11.4 miles longer. The great circle bows north of the rhumb line, which is why it starts more northerly and ends more southerly.

How to use it

1. Set the start point in Origin, by search, by clicking the map, or with Use GPS.
2. Set the end point in Destination or with a second click.
3. Read the three bearings in degrees from true north. The initial bearing also shows a 16-point compass label such as WNW.
4. The card also lists the distance, so you can see how far each heading carries you. Magnetic declination is not applied; the card says so.

The initial bearing formula

With latitudes φ1, φ2 and the longitude difference Δλ, the initial bearing is θ = atan2(sin Δλ · cos φ2, cos φ1 · sin φ2 − sin φ1 · cos φ2 · cos Δλ). The result is converted to degrees and normalized to 0 to 360, where 0 is north and 90 is east.

The final bearing is the reverse initial bearing from the second point back to the first, turned around by 180°. The bearing guide works these through by hand.

When to use the rhumb bearing

A rhumb line crosses every meridian at the same angle, so it is a straight line on a Mercator chart. Sailors and pilots used it because a single compass heading is easy to hold. Over short distances it is almost as short as the great circle. Over long east-west legs at high latitude, the extra distance becomes large.

To see both paths drawn on a globe and on Mercator, use the great circle route map.

True north, not magnetic

Every bearing on this page is measured from true north, the direction of the geographic pole. A magnetic compass points to magnetic north, which differs by the local magnetic declination. That value changes with place and year, and this tool does not ship a magnetic model, so it does not convert. Apply the declination from an official source before using these bearings with a compass.