What is a bearing in the context of Cambridge IGCSE Mathematics?

In Cambridge IGCSE Mathematics, a bearing is a way of describing the direction of one point from another using angles. Bearings are measured in degrees, clockwise from the north direction, and are typically expressed as three-digit numbers. For example, a bearing of 045° indicates a direction 45 degrees clockwise from north.

How do you calculate the bearing between two points?

To calculate the bearing between two points, first identify the north direction at the starting point. Then, measure the angle clockwise from north to the line connecting the two points. Ensure the angle is expressed as a three-digit number. For example, if the angle is 60 degrees, the bearing is written as 060°.

Why are bearings always expressed as three-digit numbers?

Bearings are expressed as three-digit numbers to maintain consistency and avoid confusion. This format ensures clarity, especially when dealing with angles less than 100 degrees. For instance, a bearing of 5 degrees is written as 005°, making it clear that it is a bearing and not just a simple angle.

What is the difference between true bearing and magnetic bearing?

True bearing is the angle measured clockwise from true north, which is the direction towards the geographic North Pole. Magnetic bearing, on the other hand, is measured from magnetic north, which is the direction a compass needle points. The difference between true north and magnetic north is known as magnetic declination, which varies depending on your location on Earth.

How are bearings used in real-life applications?

Bearings are widely used in navigation, aviation, and maritime activities to determine directions and plot courses. They are essential for pilots and ship captains to navigate accurately. In addition, bearings are used in land surveying and map reading to describe the direction from one point to another precisely.

Why is "Measuring anticlockwise instead of clockwise" a common mistake in Bearing?

Why it happens: Defaulting to maths-style positive direction. How to avoid it: Bearings: always CLOCKWISE from north. Different from standard angle convention. Source: 0580/42 — recurring

Why is "Writing a bearing with fewer than 3 digits" a common mistake in Bearing?

Why it happens: Treating it like a normal angle. How to avoid it: Pad with zeros: 30 030, 5 005.

Why is "Using the wrong north line" a common mistake in Bearing?

Why it happens: Each point has its own north line. How to avoid it: Bearing of B from A uses the north line at A, not at B.

Why is "Subtracting $180\degree$ when you should add" a common mistake in Bearing?

Why it happens: Forgetting the 0–360 range. How to avoid it: If the bearing is less than 180, ADD 180. If it's more, SUBTRACT 180.

TrigonometryCambridge IGCSE Mathematics (0580)

Bearing

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Bearings — Cambridge IGCSE 0580 Maths Extended (2026)

Compass directions written as three-figure angles measured CLOCKWISE from NORTH. Cambridge expects $0\degree \le$ bearing $< 360\degree$ — three digits even for small angles.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E8.2 — Use the sine, cosine and tangent ratios to solve problems involving bearings.

What is a bearing?

Three-figure angle, measured clockwise from north.

A bearing is a direction expressed as a three-figure angle measured clockwise from north.

Compass cardinal points.

North → $000\degree$
North-East → $045\degree$
East → $090\degree$
South-East → $135\degree$
South → $180\degree$
South-West → $225\degree$
West → $270\degree$
North-West → $315\degree$

Why three figures? Bearings always use three digits — write $060\degree$ , not $60\degree$ . This avoids confusion with non-bearing angles.

Worked. A point lies due east of an observer. The bearing is $090\degree$ .

Worked. A point lies $30\degree$ west of north. The bearing is $360\degree - 30\degree = 330\degree$ .

Clockwise from north.
Three figures always.
Range $000\degree$ to $360\degree$ .
Cardinal points: N $000$ , E $090$ , S $180$ , W $270$ .

Bearing of B from A

The phrase "bearing of B FROM A" — start at A, draw a north line, measure the angle clockwise to AB.

"Bearing of B from A" means: stand at $A$ , face north, then turn clockwise until you face $B$ .

Step-by-step.

Mark point $A$ and draw a vertical line UP from $A$ (representing north).
Draw the line from $A$ to $B$ .
Measure the clockwise angle from the north line to $AB$ .
Write as a three-figure bearing.

Worked. $A$ is at the origin, $B$ is at $(4, 3)$ (i.e. east-then-north of $A$ ). Find the bearing of $B$ from $A$ .

The angle from north (the positive $y$ -axis) clockwise to $AB$ .
$\tan(\text{angle from north}) = \dfrac{4}{3}$ (east displacement over north displacement).
Angle $= \tan^{-1}(4/3) \approx 53.13\degree$ .
Bearing $= 053\degree$ .

Tip. Drawing the north arrow at the STARTING point — and ONLY at the starting point — is the key visual.

Bearing of B from A: north arrow at A, clockwise to AB.
Express as a three-figure bearing.
Use right-angled trig (or sine/cosine rules) to compute the angle.

Back bearings

The back bearing of B from A is the bearing of A from B. Add $180\degree$ if the original is $< 180\degree$ , otherwise subtract $180\degree$ .

If the bearing of B from A is $\theta$ , the bearing of A from B is the back bearing: $\text{back bearing} = \begin{cases} \theta + 180\degree & \text{if } \theta < 180\degree \\ \theta - 180\degree & \text{if } \theta \ge 180\degree \end{cases}$

The two opposite directions are exactly $180\degree$ apart, but you stay in the $0$ to $360$ range.

Worked. Bearing of $B$ from $A$ is $075\degree$ . Find the bearing of $A$ from $B$ .

$075 + 180 = 255\degree$ . Back bearing $= 255\degree$ .

Worked. Bearing of $C$ from $D$ is $230\degree$ . Find the bearing of $D$ from $C$ .

$230 - 180 = 050\degree$ . Back bearing $= 050\degree$ .

Back bearing differs by $180\degree$ .
Add $180$ if original $< 180\degree$ .
Subtract $180$ if original $\ge 180\degree$ .
Stay within $000\degree$ to $360\degree$ .

Bearings combined with trigonometry

Find the angles inside the triangle, then apply right-angled trig or the sine/cosine rule.

Cambridge's hardest bearing questions combine a journey (multiple legs, each with a bearing and distance) with trigonometry to find a missing distance or angle.

Worked. A ship sails on a bearing of $060\degree$ for $40\,\text{km}$ , then on a bearing of $150\degree$ for $30\,\text{km}$ . How far is it from the start, and on what bearing?

Step 1. Draw the journey. Note that the two legs differ in bearing by $150 - 60 = 90\degree$ , so the angle between the two legs at the turning point is $90\degree$ (the journey makes a right angle).

Step 2. The displacement from start to end is the hypotenuse of a right-angled triangle:

$d = \sqrt{40^{2} + 30^{2}} = \sqrt{2500} = 50\,\text{km}$ .

Step 3. Find the bearing. The angle inside the triangle at the start, opposite the second leg (length $30$ ), is $\theta = \tan^{-1}(30/40) \approx 36.87\degree$ . The bearing from start to end is $060\degree + 36.87\degree \approx 097\degree$ .

General template.

Draw all north arrows at each waypoint.
Mark each bearing.
Use angle facts (alternate angles, co-interior, etc.) to find the interior angle of the triangle formed.
Apply right-angled trig (if a $90\degree$ pops out) or sine/cosine rule (general case — see those study notes).

Draw north arrows at each waypoint.
Identify the interior angle of any triangle that forms.
Apply right-angled trig, sine rule, or cosine rule.
Express the final answer as a three-figure bearing.

How it’s examined

Bearings appear most years on Paper 4 as a 4-6 mark question, often combining bearings with trigonometry (sine/cosine rule) to find distances or angles in a journey. Examiner reports flag two errors: (i) missing leading zeros in three-figure bearings, (ii) measuring anticlockwise from north (or from the wrong axis).

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E8.2); 0580/42 Oct/Nov 2024 — Q14 (bearings + cosine rule); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

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Step-by-step worked examples — Bearing

Step-by-step solutions to past-paper-style questions on bearing, written exactly the way a tutor would explain them at the board.

1Read a bearing from a diagram
Core• read bearing
Question
From point $A$ , point $B$ lies $40\degree$ east of north. Write the bearing of $B$ from $A$ .
Step-by-step solution
1. Step 1
  Bearings are measured CLOCKWISE from north and given as 3 digits.
  $\text{Bearing} = 040\degree$
Answer
$040\degree$
Examiner tip
Always state bearings as THREE digits (e.g. $040\degree$ not $40\degree$ ).
2Find a back bearing
Extended• Adapted from 0580/42 May/Jun 2024 Q14• back bearing
Question
The bearing of $B$ from $A$ is $075\degree$ . Find the bearing of $A$ from $B$ .
Step-by-step solution
1. Step 1
  Add $180\degree$ if the original bearing is less than $180\degree$ .
  $075\degree + 180\degree = 255\degree$
Answer
$255\degree$
3Combine bearings with trigonometry
Extended• bearings + trig
Question
A ship sails $20$ km from $A$ to $B$ on a bearing of $060\degree$ . Find how far east of $A$ the ship is now.
Step-by-step solution
1. Step 1
  East = adjacent leg if we measure $60\degree$ from north (so it's the side opposite the angle from north).
2. Step 2
  Use $\sin 60\degree = \dfrac{\text{east}}{20}$ .
  $\text{east} = 20 \sin 60\degree \approx 17.3\,\text{km}$
Answer
$\approx 17.3\,\text{km}$
4Bearing between three points
Extended• three-point bearing
Question
From $A$ , the bearing of $B$ is $080\degree$ . From $A$ , the bearing of $C$ is $140\degree$ . Find $\angle BAC$ .
Step-by-step solution
1. Step 1
  Both bearings share the same north line at $A$ , so subtract.
  $\angle BAC = 140\degree - 80\degree = 60\degree$
Answer
$60\degree$

Key Formulae — Bearing

The formulae you need to memorise for bearing on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Bearing convention
$\text{Measured clockwise from NORTH, given as 3 digits } 000\degree\text{–}360\degree$
When to use
Always.
Back bearing
$\text{back} = \theta \pm 180\degree \quad (+\,180\degree \text{ if } \theta < 180\degree)$
When to use
To reverse a bearing.

Key Definitions and Keywords — Bearing

Definitions to memorise and the exact keywords mark schemes credit for bearing answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Bearing
Examiner keyword
An angle measured clockwise from north, written as a 3-digit number between $000\degree$ and $360\degree$ .
Back bearing
Examiner keyword
The bearing in the reverse direction. Differs from the original by exactly $180\degree$ .
North line
Examiner keyword
A vertical line drawn at each point in a bearings diagram, used as the reference direction.

Common Mistakes and Misconceptions — Bearing

The traps other students keep falling into on bearing questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Measuring anticlockwise instead of clockwise
0580/42 — recurring
Why it happens
Defaulting to maths-style positive direction.
How to avoid it
Bearings: always CLOCKWISE from north. Different from standard angle convention.
✕Writing a bearing with fewer than 3 digits
Why it happens
Treating it like a normal angle.
How to avoid it
Pad with zeros: $30\degree \to 030\degree$ , $5\degree \to 005\degree$ .
✕Using the wrong north line
Why it happens
Each point has its own north line.
How to avoid it
Bearing of $B$ from $A$ uses the north line at $A$ , not at $B$ .
✕Subtracting $180\degree$ when you should add
Why it happens
Forgetting the $0$ – $360$ range.
How to avoid it
If the bearing is less than $180\degree$ , ADD $180\degree$ . If it's more, SUBTRACT $180\degree$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Bearing — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

TrigonometryCambridge IGCSE Mathematics (0580)

Bearing

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Bearing, ready to print or save offline.

Bearings — Cambridge IGCSE 0580 Maths Extended (2026)

Compass directions written as three-figure angles measured CLOCKWISE from NORTH. Cambridge expects $0\degree \le$ bearing $< 360\degree$ — three digits even for small angles.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E8.2 — Use the sine, cosine and tangent ratios to solve problems involving bearings.

What is a bearing?

Three-figure angle, measured clockwise from north.

A bearing is a direction expressed as a three-figure angle measured clockwise from north.

Compass cardinal points.

North → $000\degree$
North-East → $045\degree$
East → $090\degree$
South-East → $135\degree$
South → $180\degree$
South-West → $225\degree$
West → $270\degree$
North-West → $315\degree$

Why three figures? Bearings always use three digits — write $060\degree$ , not $60\degree$ . This avoids confusion with non-bearing angles.

Worked. A point lies due east of an observer. The bearing is $090\degree$ .

Worked. A point lies $30\degree$ west of north. The bearing is $360\degree - 30\degree = 330\degree$ .

Clockwise from north.
Three figures always.
Range $000\degree$ to $360\degree$ .
Cardinal points: N $000$ , E $090$ , S $180$ , W $270$ .

Bearing of B from A

The phrase "bearing of B FROM A" — start at A, draw a north line, measure the angle clockwise to AB.

"Bearing of B from A" means: stand at $A$ , face north, then turn clockwise until you face $B$ .

Step-by-step.

Mark point $A$ and draw a vertical line UP from $A$ (representing north).
Draw the line from $A$ to $B$ .
Measure the clockwise angle from the north line to $AB$ .
Write as a three-figure bearing.

Worked. $A$ is at the origin, $B$ is at $(4, 3)$ (i.e. east-then-north of $A$ ). Find the bearing of $B$ from $A$ .

The angle from north (the positive $y$ -axis) clockwise to $AB$ .
$\tan(\text{angle from north}) = \dfrac{4}{3}$ (east displacement over north displacement).
Angle $= \tan^{-1}(4/3) \approx 53.13\degree$ .
Bearing $= 053\degree$ .

Tip. Drawing the north arrow at the STARTING point — and ONLY at the starting point — is the key visual.

Bearing of B from A: north arrow at A, clockwise to AB.
Express as a three-figure bearing.
Use right-angled trig (or sine/cosine rules) to compute the angle.

Back bearings

The back bearing of B from A is the bearing of A from B. Add $180\degree$ if the original is $< 180\degree$ , otherwise subtract $180\degree$ .

The two opposite directions are exactly $180\degree$ apart, but you stay in the $0$ to $360$ range.

Worked. Bearing of $B$ from $A$ is $075\degree$ . Find the bearing of $A$ from $B$ .

$075 + 180 = 255\degree$ . Back bearing $= 255\degree$ .

Worked. Bearing of $C$ from $D$ is $230\degree$ . Find the bearing of $D$ from $C$ .

$230 - 180 = 050\degree$ . Back bearing $= 050\degree$ .

Back bearing differs by $180\degree$ .
Add $180$ if original $< 180\degree$ .
Subtract $180$ if original $\ge 180\degree$ .
Stay within $000\degree$ to $360\degree$ .

Bearings combined with trigonometry

Find the angles inside the triangle, then apply right-angled trig or the sine/cosine rule.

Cambridge's hardest bearing questions combine a journey (multiple legs, each with a bearing and distance) with trigonometry to find a missing distance or angle.

Worked. A ship sails on a bearing of $060\degree$ for $40\,\text{km}$ , then on a bearing of $150\degree$ for $30\,\text{km}$ . How far is it from the start, and on what bearing?

Step 2. The displacement from start to end is the hypotenuse of a right-angled triangle:

$d = \sqrt{40^{2} + 30^{2}} = \sqrt{2500} = 50\,\text{km}$ .

General template.

Draw all north arrows at each waypoint.
Mark each bearing.
Use angle facts (alternate angles, co-interior, etc.) to find the interior angle of the triangle formed.
Apply right-angled trig (if a $90\degree$ pops out) or sine/cosine rule (general case — see those study notes).

Draw north arrows at each waypoint.
Identify the interior angle of any triangle that forms.
Apply right-angled trig, sine rule, or cosine rule.
Express the final answer as a three-figure bearing.

How it’s examined

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E8.2); 0580/42 Oct/Nov 2024 — Q14 (bearings + cosine rule); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Bearing, ready to print or save as PDF.

Step-by-step worked examples — Bearing

Step-by-step solutions to past-paper-style questions on bearing, written exactly the way a tutor would explain them at the board.

1Read a bearing from a diagram
Core• read bearing
Question
From point $A$ , point $B$ lies $40\degree$ east of north. Write the bearing of $B$ from $A$ .
Step-by-step solution
1. Step 1
  Bearings are measured CLOCKWISE from north and given as 3 digits.
  $\text{Bearing} = 040\degree$
Answer
$040\degree$
Examiner tip
Always state bearings as THREE digits (e.g. $040\degree$ not $40\degree$ ).
2Find a back bearing
Extended• Adapted from 0580/42 May/Jun 2024 Q14• back bearing
Question
The bearing of $B$ from $A$ is $075\degree$ . Find the bearing of $A$ from $B$ .
Step-by-step solution
1. Step 1
  Add $180\degree$ if the original bearing is less than $180\degree$ .
  $075\degree + 180\degree = 255\degree$
Answer
$255\degree$
3Combine bearings with trigonometry
Extended• bearings + trig
Question
A ship sails $20$ km from $A$ to $B$ on a bearing of $060\degree$ . Find how far east of $A$ the ship is now.
Step-by-step solution
1. Step 1
  East = adjacent leg if we measure $60\degree$ from north (so it's the side opposite the angle from north).
2. Step 2
  Use $\sin 60\degree = \dfrac{\text{east}}{20}$ .
  $\text{east} = 20 \sin 60\degree \approx 17.3\,\text{km}$
Answer
$\approx 17.3\,\text{km}$
4Bearing between three points
Extended• three-point bearing
Question
From $A$ , the bearing of $B$ is $080\degree$ . From $A$ , the bearing of $C$ is $140\degree$ . Find $\angle BAC$ .
Step-by-step solution
1. Step 1
  Both bearings share the same north line at $A$ , so subtract.
  $\angle BAC = 140\degree - 80\degree = 60\degree$
Answer
$60\degree$

Key Formulae — Bearing

The formulae you need to memorise for bearing on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Bearing convention
$\text{Measured clockwise from NORTH, given as 3 digits } 000\degree\text{–}360\degree$
When to use
Always.
Back bearing
$\text{back} = \theta \pm 180\degree \quad (+\,180\degree \text{ if } \theta < 180\degree)$
When to use
To reverse a bearing.

Key Definitions and Keywords — Bearing

Definitions to memorise and the exact keywords mark schemes credit for bearing answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Bearing
Examiner keyword
An angle measured clockwise from north, written as a 3-digit number between $000\degree$ and $360\degree$ .
Back bearing
Examiner keyword
The bearing in the reverse direction. Differs from the original by exactly $180\degree$ .
North line
Examiner keyword
A vertical line drawn at each point in a bearings diagram, used as the reference direction.

Common Mistakes and Misconceptions — Bearing

The traps other students keep falling into on bearing questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Measuring anticlockwise instead of clockwise
0580/42 — recurring
Why it happens
Defaulting to maths-style positive direction.
How to avoid it
Bearings: always CLOCKWISE from north. Different from standard angle convention.
✕Writing a bearing with fewer than 3 digits
Why it happens
Treating it like a normal angle.
How to avoid it
Pad with zeros: $30\degree \to 030\degree$ , $5\degree \to 005\degree$ .
✕Using the wrong north line
Why it happens
Each point has its own north line.
How to avoid it
Bearing of $B$ from $A$ uses the north line at $A$ , not at $B$ .
✕Subtracting $180\degree$ when you should add
Why it happens
Forgetting the $0$ – $360$ range.
How to avoid it
If the bearing is less than $180\degree$ , ADD $180\degree$ . If it's more, SUBTRACT $180\degree$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Bearing — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Bearing

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Read a bearing from a diagram

2Find a back bearing

3Combine bearings with trigonometry

4Bearing between three points

Bearing convention

Back bearing

Bearing

Back bearing

North line

✕Measuring anticlockwise instead of clockwise

✕Writing a bearing with fewer than 3 digits

✕Using the wrong north line

✕Subtracting 180°180\degree180° when you should add

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Bearing — frequently asked questions

Bearing

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Read a bearing from a diagram

2Find a back bearing

3Combine bearings with trigonometry

4Bearing between three points

Bearing convention

Back bearing

Bearing

Back bearing

North line

✕Measuring anticlockwise instead of clockwise

✕Writing a bearing with fewer than 3 digits

✕Using the wrong north line

✕Subtracting 180°180\degree180° when you should add

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Bearing — frequently asked questions

✕Subtracting $180\degree$ when you should add

✕Subtracting $180\degree$ when you should add