How To Calculate Azimuth

Have you ever seen a U.S. Geological Survey topographical map, or any map heavy in terrain and other natural features as well as human-constructed elements like roads and dams? You may have noticed amid all the color and visual chaos that such maps have a grid of vertical and horizontal lines at regular intervals.

These grid lines on "topo maps" are especially helpful when you need to determine an azimuth, or more specifically, convert the numerical information contained in an azimuth into operational instructions on the ground. An azimuth is often used to plot immediate movements over land, but the term has other applications in geoscience as well.

According to the U.S. Army's definition, an azimuth is the angle between a line pointing toward the North Pole and a second line aimed at the location of interest, such as a hilltop, cell phone tower or just a set of numerical coordinates.

Imagine two lines starting as one and pointing at the top of a circle; if one stays fixed while the other traces out a full circle (like one of the "hands" of a clock), the angle between them at any instant represents the azimuth of the moving line. This can vary from 0° (degrees) to 360° (these actually represent the same azimuth, due north).

Therefore, an azimuth of 90° corresponds to a quarter of the way clockwise from 0° or 360°, which is east. Similarly, 180° is south, and 270° is west. You can get azimuths corresponding to NE, SE, SW and NW by adding or subtracting 45° to the appropriate N, E, S or W azimuth.

No complex azimuth formula is required to execute skillful land navigation. If you have an appropriately detailed map, a protractor, a pencil and precise knowledge of your starting point and desired end point (or direction), you are ready to draw an azimuth.

For example, say you are handed a map on which two points have been labeled A and B. A is to be your starting point, while B is your desired end point, which appears to be some distance south and west of A. How do you find the azimuth?

First, use a straight edge of the protractor and your pencil to draw a line through A and B. The line does not have to stop at each point; in fact, you should extend it as far as necessary to ensure that it crosses a vertical (north-south) grid line. You are now ready to reposition the protractor to determine the azimuth.

At this point, you need to remember that the angle you are looking for is not necessarily the smallest angle between the vertical grid line and the line you have drawn between A and B. In fact, this is only the case when the angle is less than 90° (i.e., between north and east). Otherwise, you have to refer to the relative positions of A and B.

In the present example, if you measure the angle between the vertical and your line, since it points between south and west, it will have a value between 0° and 90°. Say it is 30°. This would make the azimuth (180 + 30) = 210°, which makes geographical sense when you know generally what directions azimuth ranges necessarily imply.

In an example in which the A to B direction is between 90° and 180°, the shortest angle between the vertical and this line would need to be subtracted from 180° to get the correct azimuth. You can always check your answers against physical directions on the map to assure yourself they make sense.

You can find a variety of handy ways to calculate azimuths at various points on Earth in the Resources.