Bearings, explained with examples

By The STEM-Tutor team · Updated October 2026

A bearing describes a direction as an angle measured clockwise from north. Bearings are used in navigation by ships, aircraft and walkers, and they appear regularly on GCSE Maths papers, from simple measuring to multi-step trigonometry.

The three rules

1. Measure from north. Start facing north at the point you are measuring from.
2. Turn clockwise. Measure the angle turning clockwise until you face the direction you want.
3. Use three figures. Write every bearing with three digits: 45° is written 045°, and 8° is 008°.

The key phrase is “the bearing of B from A”. The word after “from” tells you where to draw the north line and put your protractor.

Compass directions as bearings

• North = 000°, east = 090°, south = 180°, west = 270°.
• North-east = 045°, south-east = 135°, south-west = 225°, north-west = 315°.

Measuring and drawing bearings

To measure a bearing, draw a north line at the starting point if there is not one, place the centre of your protractor on the point with 0° along the north line, and read the angle clockwise. For bearings over 180°, either measure the anticlockwise angle and subtract it from 360°, or measure the angle past south and add 180°. To draw a bearing, reverse the process: mark the angle clockwise from north and draw a line through the mark.

Back bearings

The bearing of A from B is called the back bearing. It points in exactly the opposite direction, so it differs by 180°:

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

This works because north lines at A and B are parallel, so co-interior angles between them add to 180°. A quick check: a bearing and its back bearing always differ by exactly 180°.

Worked examples

Example 1: a back bearing

The bearing of B from A is 120°. Find the bearing of A from B.

• 120° is less than 180°, so add 180°.
• 300°.

Example 2: another back bearing

The bearing of Q from P is 250°. Find the bearing of P from Q.

• 250° is more than 180°, so subtract 180°.
• 070°, written with three figures.

Example 3: using angle facts

A ship sails from port P on a bearing of 060° to point Q, then turns and sails on a bearing of 150° to point R. Find the angle PQR.

• At Q, the back bearing to P is 060° + 180° = 240°. The bearing to R is 150°.
• Angle PQR = 240° − 150° = 90°. A right angle means Pythagoras or SOHCAHTOA can be used next.

Example 4: Pythagoras

Using Example 3, PQ = 8 km and QR = 6 km. Find the distance PR.

• Angle PQR = 90°, so PR² = 8² + 6² = 100.
• PR = 10 km.

Example 5: finding a bearing with trigonometry

A walker goes 5 km due east, then 3 km due south. Find the bearing of the finish from the start.

• Right-angled triangle: the angle south of east is tan⁻¹(3/5) ≈ 31.0°.
• Bearing = 90° + 31.0° = 121°, to the nearest degree.

Example 6: the cosine rule (Higher)

A plane flies 200 km on a bearing of 040°, then 150 km on a bearing of 130°. Find its distance from the start.

• At the turning point, the back bearing to the start is 220°, and the new bearing is 130°, so the angle between the two legs is 220° − 130° = 90°.
• Distance = √(200² + 150²) = 250 km. With a non-right angle you would use the cosine rule instead.

Example 7: a non-right angle

A boat sails 10 km on a bearing of 070°, then 7 km on a bearing of 190°. Find its distance from the start.

• At the turn, the back bearing is 250°. The angle between the legs is 250° − 190° = 60°.
• d² = 10² + 7² − 2 × 10 × 7 × cos 60° = 149 − 70 = 79, so d ≈ 8.9 km.

Bearings and scale drawings

Many Foundation and Higher questions combine bearings with a scale drawing, such as a map with a scale of 1 cm to 2 km. To find a position, draw a north line, measure the bearing with a protractor, draw a line in that direction, and measure the correct length along it using the scale. To read a distance from a map, measure in centimetres and multiply by the scale. Use a sharp pencil and draw accurately: mark schemes usually allow only ±2° for angles and ±2 mm for lengths.

Example 8: a scale drawing

On a map with scale 1 cm : 5 km, a lighthouse is 4.6 cm from a harbour on a bearing of 072°. How far is the lighthouse from the harbour, and what is the bearing of the harbour from the lighthouse?

• Distance = 4.6 × 5 = 23 km.
• Back bearing = 072° + 180° = 252°.

Tips for bearings questions

• Always draw a clear sketch with a north line at every point where you need a bearing.
• Use parallel-line angle facts: north lines are parallel, so alternate angles are equal and co-interior angles add to 180°.
• Label the angles you find on the sketch as you go; many marks are for these intermediate angles.
• Check your answer is sensible: a point to the south-east must have a bearing between 090° and 180°.

Common mistakes

• Measuring anticlockwise instead of clockwise.
• Measuring from the wrong point: the bearing of B from A starts at A.
• Writing two figures, such as 45° instead of 045°.
• Getting the back bearing wrong by adding 180° to a bearing already over 180°.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are bearings on the Foundation paper?

Yes. Measuring, drawing and simple back bearings are on both tiers. Bearings combined with trigonometry or the sine and cosine rules are mainly Higher.

Why are bearings measured clockwise from north?

It is the convention used in navigation, so that every direction has one unique three-figure number from 000° to 359°.

Key takeaways

• Bearings are measured from north, clockwise, with three figures.
• “From” tells you where to start measuring.
• Back bearing: add or subtract 180°.
• North lines are parallel, so use parallel-line angle facts.
• Combine bearings with Pythagoras, trigonometry or the cosine rule to find distances.

Related guides

• Sine and cosine rules
• Pythagoras' theorem
• GCSE Maths trigonometry