Ever wondered how pilots flying into Lagos give directions without saying “left” or “right”?

Left and right change depending on which way you face, but north never does. Pilots, sailors and surveyors describe every direction as an angle turned clockwise from north — a bearing.

📒 Bearings and Distances: lesson notes

Give directions as three-figure bearings, find back bearings, and work out distances with Pythagoras and trigonometry.

Left and right change depending on which way you face, but north never does. Pilots, sailors and surveyors describe every direction as an angle turned clockwise from north — a bearing.

Three-figure bearings

A bearing is an angle measured from north, turning clockwise, written with three figures.

East is 090°, south is 180° and west is 270°. A direction 40° east of north is 040° (not 40°).

Compass directions convert like this: N40°E = 040°, S30°E = 180° − 30° = 150°, S20°W = 180° + 20° = 200°, N70°W = 360° − 70° = 290°.

🔑 Key idea: Bearing = angle from north, clockwise, always three figures.

Back bearings

If the bearing of B from A is 065°, what is the bearing of A from B? You are now facing the opposite way, which is a half-turn: 065° + 180° = 245°.

If the bearing is more than 180°, subtract instead: the back bearing of 300° is 300° − 180° = 120°.

This works because the north lines at A and B are parallel.

🔑 Key idea: Back bearing = bearing ± 180°.

Finding distances

Due north/south and due east/west are at right angles. So 6 km north then 8 km east puts you √(6² + 8²) = √100 = 10 km from the start (Pythagoras).

If you travel d km on a bearing θ between 000° and 090°, draw a right-angled triangle with the path as hypotenuse. The angle θ is at the start, measured from north.

Distance north = d cos θ (adjacent) and distance east = d sin θ (opposite). For 10 km on 030°: east = 10 × sin 30° = 5 km, north = 10 × cos 30° ≈ 8.7 km.

🔑 Key idea: Sketch first. Right angle → Pythagoras; angle from north → north = d cos θ, east = d sin θ.

Worked example

Musa walks 30 km due north and then 16 km due east. How far is he from the starting point, in a straight line?

Step 1: Right-angled triangle
The two legs are 30 km and 16 km; the distance d is the hypotenuse.
Step 2: Pythagoras
d² = 30² + 16² = 900 + 256 = 1156
d = √1156 = 34 km

Answer: 34 km

Summary

Bearings are measured clockwise from north and written with three figures.
Back bearing: add 180° if the bearing is less than 180°, otherwise subtract 180°.
North–south and east–west paths meet at right angles: use Pythagoras.
On a bearing θ: north = d cos θ, east = d sin θ.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.