Bearings
A bearing describes a direction. There are only three rules, but each one is a mark, and the errors recorded in lessons were almost entirely about which point you measure from and which way round you turn.
1. The three rules
1. Measure from NORTH 2. Measure CLOCKWISE 3. Always write THREE FIGURES
Three figures means adding leading zeros:
| Direction | Bearing |
|---|---|
| North | 000° |
| East | 090° |
| South | 180° |
| West | 270° |
| North-east | 045° |
Write 045°, not 45°. The three-figure form is a mark in its own right, and tutors flagged it repeatedly. A bearing is always between 000° and 360°.
Clockwise from north — always. Measuring anticlockwise, or from the south line, produces exactly the kind of error recorded in lessons: 129° instead of 231°. If your answer and the correct one add to 360, you measured the wrong way round.
Read the digits carefully. 303° and 330° are different directions and were confused in a real lesson — check what you’ve written.
2. “The bearing of B from A”
This phrase decides everything, and getting it backwards is the most common error in the topic.
“The bearing of B FROM A” means: stand at A, and measure the direction to B.
Method:
- Put your north line at A — the point after the word “from”
- Turn clockwise from that north line until you face B
- Read the angle, in three figures
The north line always goes at the “from” point. Drawing it at the wrong point gives a completely different answer.
A memory hook: the point after “from” is where you stand; the other point is where you look.
3. Back bearings
The bearing of A from B is the reverse of the bearing of B from A.
If the bearing is less than 180°, ADD 180°. If it is 180° or more, SUBTRACT 180°.
| Bearing of B from A | Bearing of A from B |
|---|---|
| 070° | 250° |
| 120° | 300° |
| 250° | 070° |
| 310° | 130° |
The two bearings always differ by exactly 180° — because you’re facing the opposite way. The rule about adding or subtracting simply keeps the answer inside 000°–360°.
This is where the “215 instead of 250” type of error comes from: the reverse of 070° is 070 + 180 = 250°, not something read off the diagram by eye.
4. Using parallel north lines
Every north line on a diagram points the same way, so all north lines are parallel — which unlocks the angle rules from geometry.
Co-interior angles between two parallel north lines add to 180°.
That single fact is what makes the back-bearing rule work, and it is how most multi-stage bearing questions are solved.
Example: the bearing of B from A is 070°. Find the bearing of A from B.
- At B, draw the north line
- The angle between the north line at B and the line BA is co-interior with 70°, so it is 180 − 70 = 110°
- Measuring clockwise from north at B: 360 − 110 = 250° ✓
Alternate and corresponding angles apply too. Once you draw both north lines, a bearings question becomes an ordinary parallel-lines problem.
5. Bearings with trigonometry
Harder questions give distances and ask for a bearing, or vice versa.
Method:
- Draw a clear diagram, with north lines at every relevant point
- Mark all given distances and angles
- Find the angles inside the triangle using the bearings and angle rules
- Use Pythagoras, SOHCAHTOA, or the sine/cosine rule as appropriate
- Convert your triangle angle back into a bearing — measuring clockwise from north
Step 5 is the one that gets skipped. An angle inside the triangle is not the bearing. You must relate it back to the north line before writing your answer.
Draw the diagram even when one is given, if you need to add north lines. Tutors repeated this more often here than anywhere else in the syllabus — bearings questions are unsolvable without a clear diagram, and marks are given for the diagram itself.
Label every angle and distance you find on the diagram. Examiners award marks for these steps.
For scale drawings: use a ruler and a protractor, measure carefully, and state the units — a distance in centimetres on the drawing must be converted using the scale.
Speed, distance and time often appear alongside:
distance = speed × time — and check the time units. 30 minutes is 0.5 hours, not 30, when speed is in km/h. Confusing speed with distance was a documented error.
6. Mistakes that cost marks
Writing a bearing with two digits, e.g. 45° instead of 045°.
Measuring anticlockwise instead of clockwise.
Putting the north line at the wrong point — it goes at the “from” point.
Reversing “the bearing of B from A”.
Adding 180° when you should subtract, or the reverse.
Giving a triangle angle as the bearing.
Not drawing north lines at each point.
Failing to label angles and distances.
Mixing time units in speed calculations.
Omitting units on distances.
Frequently asked questions
What is a bearing? A direction measured clockwise from north, written with three figures.
Why three figures? It is the convention, and it prevents ambiguity — 045° rather than 45°.
What does “the bearing of B from A” mean? Stand at A, put your north line there, and measure clockwise to B.
Where does the north line go? At the point after the word “from”.
What is a back bearing? The reverse direction — add 180° if the bearing is under 180°, otherwise subtract 180°.
What is the bearing of A from B if B from A is 070°? 250°.
Why are north lines useful? They are all parallel, so alternate and co-interior angle rules apply.
How do I find a bearing using trigonometry? Find the triangle angle with SOHCAHTOA or the sine/cosine rule, then convert it to a bearing clockwise from north.
What’s the bearing due west? 270°.
Quick revision checklist
- I measure from north, clockwise, in three figures
- I know the bearings of the four compass points
- I put the north line at the “from” point
- I can interpret “the bearing of B from A” correctly
- I can find a back bearing by adding or subtracting 180°
- I know the two bearings differ by exactly 180°
- I use parallel north lines with co-interior and alternate angles
- I draw a clear diagram with all north lines
- I label every angle and distance
- I convert triangle angles back into bearings
- I can combine bearings with Pythagoras and trigonometry
- I use a ruler and protractor for scale drawings
- I check my time units in speed calculations
These notes cover bearings in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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