Bearings and Navigation

Describing direction precisely

"Head north-ish, then turn a bit right" is useless to a pilot or sailor. Bearings give direction as an exact angle, so any two people describe the same course identically. They are the language of navigation for ships, aircraft, hikers and surveyors.

What a bearing is

A bearing is an angle measured:

• Clockwise, and
• from north, and
• written with three figures (000° to 360°).

So east is 090°, south is 180°, west is 270°, and a full turn back to north is 360° (or 000°). A direction of 7° clockwise from north is written 007° — always three digits.

Direction	Bearing
North	000°
North-east	045°
East	090°
South-east	135°
South	180°
South-west	225°
West	270°
North-west	315°

Measuring a bearing

To measure or read a bearing from A to B:

1. Draw the north line (straight up) at point A — the place you are standing.
2. Measure the angle clockwise from that north line round to the line AB.
3. Write it as three figures.

The most common mistake is measuring from the wrong point. The north line always goes at the start of the journey.

Worked example 1: From a harbour H, a boat sails so that the angle clockwise from north round to the boat's direction is 110°. State the bearing.

1. The angle from north, clockwise, is 110°.
2. The bearing is 110° (already three figures).

Worked example 2: Town B is directly west of town A. What is the bearing of B from A?

1. West is a quarter-turn clockwise past south: north → east → south → west.
2. That is 270° clockwise from north.
3. Bearing = 270°.

Back bearings

A back bearing is the bearing for the return trip — from B back to A. Because you are now looking the opposite way, it differs by exactly 180°.

• If the original bearing is less than 180°, add 180°.
• If the original bearing is more than 180°, subtract 180°.

Worked example 3: The bearing of B from A is 050°. Find the bearing of A from B.

1. 050° is under 180°, so add 180°: 050° + 180° = 230°.
2. The back bearing is 230°.

Worked example 4: The bearing of a lighthouse from a ship is 215°. What is the bearing of the ship from the lighthouse?

1. 215° is over 180°, so subtract 180°: 215° − 180° = 035°.
2. The back bearing is 035°.

Combining bearings on a journey

Many problems involve two legs of a journey. The trick is to use parallel north lines and the rule that co-interior (allied) angles between parallel lines add to 180°.

Worked example 5: A ship sails on a bearing of 060° from A to B, then turns and sails on a bearing of 150° from B to C. Find the angle ABC (the angle of the turn inside the path).

1. Draw a north line at B. The bearing back to A (back bearing of 060°) is 060° + 180° = 240°.
2. The bearing on to C is 150°.
3. The interior angle ABC is the difference: 240° − 150° = 90°.
4. Angle ABC = 90° — a right angle, which makes the next calculation easy.

Bearings with trigonometry

Once a journey is sketched, the points form a triangle and you can find missing distances. Use right-angled trigonometry when there is a right angle, otherwise the sine and cosine rules.

Worked example 6: Using example 5, suppose AB = 8 km and BC = 6 km, with angle ABC = 90°. Find the direct distance AC.

1. The triangle is right-angled at B, so use Pythagoras.
2. AC² = AB² + BC² = 8² + 6² = 64 + 36 = 100.
3. AC = √100 = 10 km.

This uses the Pythagorean theorem. If the angle is not 90°, switch to the law of cosines.

Worked example 7: From A, a plane flies 120 km on a bearing of 040° to B, then 90 km on a bearing of 100° to C. Find the distance AC.

1. North line at B: back bearing to A is 040° + 180° = 220°; bearing on to C is 100°.
2. Angle ABC = 220° − 100° = 120°.
3. Cosine rule: AC² = AB² + BC² − 2·AB·BC·cos(ABC) = 120² + 90² − 2(120)(90)cos120°.
4. cos 120° = −0.5, so AC² = 14400 + 8100 − 21600(−0.5) = 22500 + 10800 = 33300.
5. AC = √33300 ≈ 182.5 km.

For non-right triangles like this, see the sine and cosine rules for the full method, including finding the final bearing of C from A.

Why measure clockwise from north? Compasses point north, so north is the one direction everyone shares. Measuring clockwise (the direction clock hands move) is a worldwide convention, so a bearing of 075° means the same thing to every navigator, on every map, in every country.

Where bearings are used

Ships and aircraft are steered by bearings, and air-traffic control uses them to keep planes apart. Hikers and orienteers take compass bearings to walk a straight line to a distant point they cannot see. Surveyors fix the positions of new buildings and boundaries with bearings, and search-and-rescue teams describe locations with them. Anywhere precise direction matters, bearings are the standard.

Practice activity

1. Write a direction of 9° clockwise from north as a proper bearing.
2. State the bearing of due south-east.
3. Find the back bearing of 075°.
4. Find the back bearing of 290°.
5. A boat sails 5 km north then 12 km east. Use Pythagoras to find its direct distance from the start.

Answers:

1. 009°.
2. South-east = 135°.
3. Under 180°, add 180° → 255°.
4. Over 180°, subtract 180° → 110°.
5. √(5² + 12²) = √(25 + 144) = √169 = 13 km.

Summary

A bearing is an angle measured clockwise from north, always written as three figures from 000° to 360°. Measure from the north line at the point you start. A back bearing reverses the trip: add 180° if under 180°, subtract 180° if over. For journeys with two legs, use parallel north lines to find the interior angle, then apply Pythagoras, the sine rule or the cosine rule to find distances and directions. Bearings are the precise, universal language of navigation.