Bearings in maths: how to measure and calculate them

1. Draw a north line at the starting point.
2. Place the protractor's zero on the north line.
3. Measure clockwise to the line joining the two points.
4. For bearings over 180°, measure the anticlockwise angle and subtract it from 360°, or use a full-circle protractor.

The bearing of $A$ from $B$ is the reverse direction: add 180° if the original is less than 180°, otherwise subtract 180°.

• The bearing of $B$ from $A$ is 060°, so the bearing of $A$ from $B$ is $060 + 180 = 240°$.
• A bearing of 300° reverses to $300 - 180 = 120°$.

This works because the two north lines are parallel, so the angles are co-interior and add up to 180° inside the parallel lines.

A walker goes 5 km on a bearing of 060°. How far east and north are they?

The angle from north is 60°, so:

• East $= 5 \sin 60° \approx 4.33$ km.
• North $= 5 \cos 60° = 2.5$ km.

For journeys in two stages, draw both legs, find the angle inside the triangle, and use the cosine rule for the distance and the sine rule for the angle.

Many bearing questions use a scale: 1 cm represents 2 km, for example. Measure the bearing with a protractor and the distance with a ruler, then multiply by the scale. Use a sharp pencil and draw the north line carefully at each point: small errors grow over long distances.

• Measuring anticlockwise, or from the east.
• Writing two figures (45° instead of 045°).
• Measuring at the wrong point: "from A" means stand at A.
• Forgetting to draw a new north line at the second point.

Bearings combine angle facts, scale drawing and trigonometry, so they're a favourite for multi-step questions. They're used in navigation, orienteering and aviation. Practise the trigonometry part with our trigonometry calculator and trigonometry questions.

1. The bearing of $B$ from $A$ is 075°. Find the bearing of $A$ from $B$.
2. The bearing of $Q$ from $P$ is 210°. Find the bearing of $P$ from $Q$.
3. A ship sails 8 km due east, then 6 km due south. How far is it from its start, and on what bearing?

Answers: 1. 255°. 2. 030°. 3. 10 km (Pythagoras), on a bearing of about 127° ($90° + \tan^{-1}(6 \div 8) \approx 126.9°$).

• Draw a north line at every point a bearing is measured from, even if the diagram doesn't show one.
• Mark the angle you're using and say which rule you used: "co-interior angles add up to 180°" for back bearings, or "angles around a point add up to 360°".
• Give bearings to the nearest degree unless told otherwise, always with three figures.