Bearings

"Head north-east-ish for a bit" is no way to sail a ship or fly a plane. Navigators, pilots and mountain-rescue teams need a direction that is exact and can't be misread — so they use a bearing: a single angle measured clockwise from North, always written with three figures. "Roughly north-east" becomes a crisp 070^\circ. Combine a bearing with a distance and some trigonometry, and "sail on this heading for this far" turns into an exact position on the map.

The rules of a bearing

A bearing describes a direction as an angle. It is always measured in degrees, clockwise from North, and written with three figures — so a small bearing carries a leading zero or two, like 045^\circ, 130^\circ or 270^\circ.

The four main compass directions are easy to read off:

North is 000^\circ;
East is 090^\circ;
South is 180^\circ;
West is 270^\circ.

Turning all the way round brings you back to North at 360^\circ, so every bearing lies between 000^\circ and 360^\circ.

From compass words to bearings

The eight-point compass slots straight onto the bearing dial — each step is 45^\circ clockwise from North:

N = 000^\circ, NE = 045^\circ, E = 090^\circ, SE = 135^\circ;
S = 180^\circ, SW = 225^\circ, W = 270^\circ, NW = 315^\circ.

So "north-east" is a friendly name for 045^\circ, and "south-west" is 225^\circ. Notice the pattern: opposite directions (N and S, NE and SW) always differ by exactly 180^\circ — which is precisely why a back bearing is the forward bearing \pm\,180^\circ.

A bearing describes a direction by an angle:

it is measured clockwise from North;
it is always written with three figures, from 000^\circ to 360^\circ;
the back bearing — the direction from B back to A, given the bearing of B from A — is that bearing \pm\,180^\circ (add 180^\circ if the bearing is under 180^\circ, subtract it if it is over);
combine bearings with right-angled trigonometry to turn directions into distances.

See a bearing

Step through the figure: a North line, a direction, and the angle swept clockwise from North to reach it.

Worked example 1 — reading a bearing off a diagram

Point B lies to the north-east of A. From A we draw the North line straight up, then the arrow to B, and measure the angle between them clockwise: it comes to 60^\circ.

Written correctly with three figures, the bearing of B from A is 060^\circ — not 60^\circ. The leading zero is part of the rule, not decoration.

Worked example 2 — back bearings

The back bearing is the return direction — if you know how to get from A to B, the back bearing tells you how to get home. It differs by exactly 180^\circ.

The bearing of B from A is 110^\circ (under 180^\circ), so add: the back bearing of A from B is 110^\circ + 180^\circ = 290^\circ.
The bearing of B from A is 250^\circ (over 180^\circ), so subtract: the back bearing is 250^\circ - 180^\circ = 070^\circ.

Add or subtract to stay inside the 000^\circ–360^\circ range.

Worked example 3 — bearings meet trigonometry

A ship sails 120\text{ km} on a bearing of 040^\circ. How far North and how far East has it travelled?

Draw the North line and the ship's track: they make an angle of 40^\circ. The northward leg lies alongside the North line (adjacent to the angle), and the eastward leg is opposite it. With the 120\text{ km} track as the hypotenuse:

\text{North} = 120\cos 40^\circ \approx 91.9\text{ km}, \qquad \text{East} = 120\sin 40^\circ \approx 77.1\text{ km}.

So the ship ends up about 92 km north and 77 km east of where it started — an exact position from a heading and a distance. This is the heart of dead reckoning: stitch several such legs together and you can plot a whole voyage. When the triangles aren't right-angled, the sine rule takes over.

Bearings punish sloppiness. Guard against these three:

Clockwise from North, three digits. Measuring anticlockwise, or from East instead of North, flips the answer entirely. And a bearing under 100^\circ must keep its leading zeros: 70^\circ is written 070^\circ, 5^\circ is 005^\circ.
"From" tells you where to stand. The bearing of A from B starts at B, looking towards A — not the other way round. The order of the words matters, and getting it backwards is the single most common bearings mistake.
Draw the North line at the STARTING point. Always plant a fresh North arrow at the point you are measuring from, then sweep clockwise. Do that and the diagram keeps you honest.

Bearings are the real language of navigation. Ships, aircraft, mountain-rescue teams and orienteers all speak in three-figure bearings — and getting one wrong is not just an exam slip. Real vessels have run aground because a bearing was read anticlockwise, or a leading zero was dropped, or "from" and "to" were swapped.

For centuries, long before satellites, sailors crossed oceans by dead reckoning: hold a known bearing for a known time at a known speed, mark the new position, then start the next leg. Bearing by bearing, trig turns each heading-and-distance into a step North and a step East, and the steps add up to a course across the whole sea. Add the sine and cosine rules for triangles that aren't right-angled, and you can calculate exactly where a multi-leg journey ends up — the maths that guided explorers home.