IGCSE / O Level Mathematics · Topic 6 Trigonometry · Chapter 28 · pp. 392–393

Bearings

A bearing is a direction written as an angle. Ships, aircraft and walkers all use it, because three digits are clearer than “a bit east of north”. This chapter shows how to read a bearing, how to find the way back, and how to draw a journey to scale.

Learning objectives

Read and write a three-figure bearing, measured clockwise from north.
Link the eight compass points to their bearings.
Find the back bearing of a journey leg using parallel north lines.
Make a scale drawing of a journey and read the distance and bearing from it.

Book exercises: Exercise 28.1 and Student assessment 1 (pp. 392–393).

Three-figure bearings

🔑 Key ideas

Sailors once used 16, later 64, compass points. Faster travel needed something more exact: the three-figure bearing.
North is 000°. You turn clockwise; one full turn is 360°.
Always write three figures: 45° is written 045°, and 7° is 007°.
“The bearing of B from A” means: stand at A, face north, turn clockwise until you face B.

The eight compass points and their bearings. Each step round the circle clockwise adds 45°.

The bearing of B from A is 135°: the angle at A, measured clockwise from the north line through A.

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

💡 “From” tells you where to stand. The bearing of B from A is measured at A. The bearing of A from B is measured at B, and it is a different number.

📘 Example 1 · Write each as a three-figure bearing: 62°, east, south west

62° has two figures, so put a zero in front: 062°.
East is a quarter turn clockwise from north: 090°.
South west is halfway between south (180°) and west (270°): (180 + 270) ÷ 2 = 225°.

📘 Example 2 · A ship sails due south. What is its bearing, and how far has it turned from north?

Start facing north (000°) and turn clockwise.
Facing south is a half turn: 360° ÷ 2 = 180°.
The bearing is 180°.

✏️ Try it

Write 7° as a three-figure bearing.
What is the bearing of north west?
A walker faces east, then turns 90° clockwise. What is the new bearing?

Answers

1. 007° · 2. 315° (midway between 270° and 360°) · 3. 090° + 90° = 180°

Next: back bearings →

Back bearings

🔑 Key ideas

If the bearing of B from A is known, the back bearing is the bearing of A from B: the way home.
The two north lines are parallel, so alternate angles are equal. That is why the two bearings differ by exactly 180°.
If the bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°.
The answer must stay between 000° and 360°, written with three figures.

The north lines at A and B are parallel. The angle p inside the zigzag equals 135° (alternate angles), so the back bearing is 180° + 135° = 315°.

Back-bearing rule. Check: your answer is always between 0° and 360°, and the two bearings differ by 180°.

📘 Example 3 · The bearing of B from A is 135°. Find the bearing of A from B (the book's example)

Draw the north line at A and at B. They are parallel.
135° is less than 180°, so add: 135° + 180° = 315°.
The bearing of A from B is 315°.
Check: 315° − 135° = 180°. ✔

📘 Example 4 · The bearing of B from A is 245°. What is the bearing of A from B?

245° is more than 180°, so subtract: 245° − 180° = 65°.
Write it with three figures: 065°.
Check: 245° − 65° = 180°. ✔

📘 Example 5 · A ship sails from port on a bearing of 300°. What bearing takes it home?

300° is more than 180°, so subtract 180°: 300° − 180° = 120°.
The way home is 120°.
Check: 120° + 180° = 300°. ✔

🧪 Try it: back-bearing compass

Bearing of B from A (0 to 360°)

A and B are drawn with a north line at each. The arc at A is the bearing of B from A; the arc at B is the back bearing.

Working

✏️ Try it

The bearing of Q from P is 060°. Find the bearing of P from Q.
The bearing of Q from P is 215°. Find the bearing of P from Q.
A boat sails on a bearing of 090°. On what bearing must it sail to return?

Answers

1. 060° + 180° = 240° · 2. 215° − 180° = 035° · 3. 090° + 180° = 270°

Next: scale drawings →

Scale drawings of journeys

🔑 Key ideas

A scale such as 1 cm : 1 km means every 1 cm on paper stands for 1 km in real life.
Take north to be a line straight up the page. Draw a fresh north line at every point you turn at.
Place the protractor centre on the point, its 0° on the north line, and count clockwise.
To find “the distance and bearing of A from C”, join C to A, then measure the length and the angle at C.

Drawing a journey: for each leg, draw north, measure the angle, then measure the scaled length. Join the ends last.

Example 6: A to B is 6 km on 060°, then B to C is 8 km on 150°. The dashed line is the answer to find.

📘 Example 6 · Scale 1 cm : 1 km. Start at A, go 6 km on 060° to B, then 8 km on 150° to C. Find the distance and bearing of A from C

Mark A. Draw a north line up the page through A.
Measure 60° clockwise from north. Draw AB 6 cm long (6 km ÷ 1 km per cm).
At B, draw a new north line, parallel to the first. Measure 150° clockwise from it.
Draw BC 8 cm long. Join C to A.
Measure CA: it is 10 cm, so the distance is 10 km.
At C, draw a north line and measure clockwise to CA: about 293°.
Check: the angle at B is 150° − 60° = 90°, so 6, 8, 10 is a right-angled triangle. ✔ Also, C from A is about 113°, and 113° + 180° = 293°. ✔

💡 A scale drawing gives answers to about ±1 mm and ±2°. Your measured answer will be close to, not exactly, the calculated one. The back bearing is a good check on your protractor work.

📘 Example 7 · Scale 1 cm : 5 km. A lighthouse is 4.6 cm from a port on the drawing. How far is it really?

1 cm stands for 5 km.
Multiply: 4.6 × 5 = 23.
The real distance is 23 km.

✏️ Try it

Scale 1 cm : 2 km. How long is a leg of 9 km on the drawing?
A ship sails 3 km on 090°, then 4 km on 180°. How far is it from its start?
For the ship in question 2, find the bearing of the start from the ship, to the nearest degree.

Answers

1. 9 ÷ 2 = 4.5 cm · 2. The legs meet at a right angle: √(3² + 4²) = 5 km · 3. The ship is 3 km east and 4 km south of the start, so the start is up-left of it. tan⁻¹(3 ÷ 4) = 37°, so the bearing is 360° − 37° = 323°

Next: summary and quiz →

Chapter summary

A bearing is measured clockwise from north and written with three figures (045°, 007°, 270°).
The bearing of B from A is measured at A.
Back bearing: if the bearing is under 180°, add 180°. Otherwise subtract 180°.
Scale drawing: draw a north line at every point, measure the angle clockwise, divide real lengths by the scale.
Read the answer bearing at the point you are asked “from”.

📄 Practice questions

The book's Exercise 28.1 and Student assessment 1 are in the printable assignment for this chapter.

Mini-quiz

1. From which direction, and which way round, are bearings measured?

2. How is a bearing of 45° written as a three-figure bearing?

3. What is the bearing of south west?

4. What is the bearing of west?

5. The bearing of B from A is 135°. What is the bearing of A from B?

6. The bearing of B from A is 245°. What is the bearing of A from B?

7. A ship sails on a bearing of 300°. What bearing takes it back?

8. A bearing and its back bearing always differ by how much?

9. The bearing of Q from P is 040°. What is the bearing of P from Q?

10. On a scale of 1 cm : 5 km, a leg measures 6 cm. How long is it really?

11. A boat goes 3 km on 090°, then 4 km on 180°. How far is it from its start?

Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 28 (Bearings). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.