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Bearings

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Cambridge (CIE) IGCSE — International Maths: Extended

Bearings

Big idea: A bearing is just a fancy compass direction — always measured from North, always clockwise, and always written as a 3-digit angle like 059° instead of 59°.

📋 Summary

Bearings describe an angle used mainly for navigation — think of it as "which direction, exactly."
Three unbreakable rules: measured from North, measured clockwise, written with 3 digits (e.g. 007°, 090°, 245°).
To find a bearing between two points: draw North at the starting point, join the two points with a line, measure the clockwise angle between them.
To plot a point on a given bearing: draw North at the start point, measure the angle clockwise, then mark the distance along that new line.
Back bearings: the bearing of B from A and the bearing of A from B differ by exactly 180°. Add 180° if the original bearing is under 180°; subtract 180° if it's over.
Trickier bearings questions often bring in Pythagoras and trigonometry (especially for right-angled triangles formed between North lines and travel paths).

🧭 What Are Bearings?

A bearing is simply a way of describing a direction as an angle. Instead of saying "go north-east-ish," a bearing lets you say exactly which way to go — down to the degree. This precision is exactly why ships, planes, and hikers use bearings instead of vague compass words.

Every single bearing you ever measure or draw must follow three rules, no exceptions:

The Three Golden Rules

1. Measured FROM North · 2. Measured CLOCKWISE · 3. Written with 3 DIGITS

In plain English: always start facing North, always turn to the right (clockwise) to find your direction, and always write the angle using three digits — so 59° becomes 059°, and 8° becomes 008°.

Why "3 digits" matters

Bearings range from 000° to 360°. Writing "59°" instead of "059°" isn't just untidy — in an exam, it can cost you a mark because it doesn't follow the agreed convention. Always pad with a zero if needed.

The Compass Directions as Bearings

It really helps to know these off by heart, since they come up constantly as reference points:

The eight main compass points and their bearings — N is always 000°/360°, and you turn clockwise from there.

Direction	Bearing
North (N)	000° (or 360°)
North-East (NE)	045°
East (E)	090°
South-East (SE)	135°
South (S)	180°
South-West (SW)	225°
West (W)	270°
North-West (NW)	315°

Practice Question

A hiker walks in a direction that is halfway between East and South. What bearing is this, written correctly?

Practice Question

Write the following as proper 3-digit bearings: (a) 7° (b) 84° (c) 5°

📐 Finding a Bearing Between Two Points

This is the classic "measure the bearing of A from B" question. The trickiest part isn't the measuring — it's knowing where to stand before you even pick up your protractor.

The wording trap

"The bearing of A from B" → you stand at B and look towards A.
"The bearing of B from A" → you stand at A and look towards B.
The word right after "from" tells you where to plant your feet and draw your North line. This single detail trips up more students than the actual angle-measuring does!

Step-by-Step Method

Identify the starting point — the one mentioned after "from."
Draw a North line at that starting point (a straight vertical arrow pointing up, always labelled N).
Draw a line connecting the starting point to the other point.
Measure the angle between the North line and this new line, going clockwise from North.
Write the answer using 3 digits.

Quick Rule

Bearing = clockwise angle from North line at the STARTING point, to the line joining the two points

Always draw North at the point you're starting from — not the destination.

Practice Question

Town Y is due East of Town X. What is the bearing of Y from X, and what is the bearing of X from Y?

✏️ Drawing a Point on a Bearing

This is the reverse skill: instead of measuring an existing angle, you're given a bearing and a distance, and asked to plot exactly where that point should go — usually on a scale drawing.

Step-by-Step Method

Step 1: Draw a North line at the point you're measuring from. (If you're told "the bearing from A to B," the North line goes at A.)
Step 2: Using a protractor, measure the given angle clockwise from the North line, and draw a straight line in that direction.
Step 3: Convert the real-world distance into the scale distance (using the map's scale), then measure that length along the line and mark point B.

Don't forget the scale conversion

If the scale is "1 cm = 10 km" and you need to plot a point 70 km away, you must first convert: 70 km ÷ 10 = 7 cm on your drawing. Students often measure the raw km value with their ruler by mistake — always convert first!

Worked Example (from the textbook)

A ship sets sail from point P. It sails on a bearing of 105° until it reaches point Q, 70 km away. It then changes path and sails on a bearing of 065° for a further 35 km. Scale: 1 cm = 10 km.

P → Q on a bearing of 105° (7 cm), then Q → final position on a bearing of 065° (3.5 cm)

Working:

Draw North at P, measure 105° clockwise, draw the line, convert 70 km → 7 cm, mark Q.
Draw a new North line at Q (North lines are always parallel to each other, but each point gets its own), measure 65° clockwise, convert 35 km → 3.5 cm, mark the final position.

Key habit

Every time the ship (or object) changes direction, you draw a brand new North line at that new point. North is always "straight up" on the page — it doesn't rotate with the direction of travel.

Practice Question

Using a scale of 1 cm = 5 km, how many cm would you measure to represent a distance of 42 km?

🔄 Back Bearings: Bearing of B from A ↔ A from B

Once you know the bearing in one direction, you can find the bearing in the opposite direction without measuring anything new — just add or subtract 180°.

Back Bearing Rule

If bearing of A from B < 180° → ADD 180° to get bearing of B from A

If bearing of A from B > 180° → SUBTRACT 180° to get bearing of B from A

Why? Because North lines at A and B are parallel, and the two bearings are related by alternate/co-interior angle facts — turning around 180° always points you back the way you came.

The North lines at A and B are parallel — this is what makes the "add or subtract 180°" shortcut work.

Practice Question

The bearing of Village B from Village A is 230°. What is the bearing of Village A from Village B?

Practice Question

The bearing of a lighthouse from a boat is 072°. What is the bearing of the boat from the lighthouse?

🧩 Trickier Bearings Questions (Pythagoras & Trigonometry)

Harder exam questions won't just ask you to measure or draw — they'll ask you to calculate missing distances or angles using right-angled triangle skills. The bearings context is really just "dressing" on top of a Pythagoras or trigonometry question underneath.

Golden habit

Always sketch a diagram if one isn't already given — even a rough one. Mark on every known angle and length. Bearings problems become far easier to solve once you can literally see the right-angled triangle hiding inside them.

How These Questions Usually Work

You're given two bearings and a distance (or two distances) and asked to find a missing length or angle.
North lines at different points are always parallel — this lets you find angles between paths using alternate angles or angle sum rules.
Once you've found the angle(s) inside your triangle, use SOHCAHTOA (for right-angled triangles) or the sine/cosine rules (for non-right-angled triangles) to find missing sides or angles.
Use Pythagoras' theorem (a² + b² = c²) whenever the triangle formed has a right angle and you need a missing side.

Tools You'll Combine With Bearings

Pythagoras: a² + b² = c² | SOHCAHTOA | Sine Rule | Cosine Rule

The bearing itself just tells you the angle to use — the actual number-crunching is standard triangle maths you already know.

Practice Question

A ship sails 8 km due North from port A to port B, then 6 km due East from B to C. What is the bearing of C from A, and how far apart are A and C?

🧠 What to Memorise

Term / Rule	Meaning
Bearing	An angle measured from North, clockwise, written as 3 digits
3-digit rule	059° not 59°; 008° not 8°; angles ≥100° stay as they are
"Bearing of A from B"	Stand at B, draw North at B, measure the angle to A
Back bearing (< 180°)	Add 180° to find the reverse bearing
Back bearing (> 180°)	Subtract 180° to find the reverse bearing
North (000°)	Straight up on every diagram — always redrawn at each new point
East / South / West	090° / 180° / 270°
Scale conversion	real distance ÷ scale factor = distance to measure on paper
Pythagoras' theorem	a² + b² = c², for right-angled triangles formed by paths

✅ Concepts Checklist

I can state the three rules of bearings (from North, clockwise, 3 digits)

I know the bearings of all 8 main compass directions (N, NE, E, SE, S, SW, W, NW)

I understand the difference between "bearing of A from B" and "bearing of B from A"

I can measure a bearing between two points using a North line and protractor

I can plot a point on a scale drawing given a bearing and a real-world distance

I remember to convert real distances into scale distances before measuring

I can find a back bearing by adding or subtracting 180° correctly

I can combine bearings with Pythagoras and SOHCAHTOA to solve harder problems

I always draw a new North line (parallel to the others) at every new point

🎯 Exam Tips & Common Mistakes

Forgetting the leading zero(s) Writing "45°" instead of "045°" is one of the most common lost marks. Bearings are always 3 digits — no exceptions, even for angles under 10°.

Drawing North at the wrong point "Bearing of X from Y" means you stand at Y. Students frequently draw the North line at X by mistake — always locate the word right after "from" first.

Forgetting to convert scale distances If a question gives real-world km but the drawing uses cm, always divide by the scale factor before you measure with your ruler.

Measuring anticlockwise by accident Protractors can be read two ways. Always double-check you're measuring clockwise from North, not the shorter route round.

Draw a diagram — every time Even for calculation-only questions with no map given, a rough sketch with North lines and marked angles will help you spot the right-angled triangle and avoid careless errors.

Bring the right equipment A sharp pencil, rubber, ruler, and protractor with clearly readable markings are essential — bearings questions are marked on accuracy, and a blunt pencil or worn-out protractor can cost you real marks.

Using the wrong 180° operation for back bearings Remember: bearing under 180° → ADD 180°. Bearing over 180° → SUBTRACT 180°. Mixing these up gives an answer outside the 000°–360° range, which is your clue something's gone wrong.

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Also in the full note

🧩 Trickier Bearings Questions (Pythagoras & Trigonometry)
🎯 Exam Tips & Common Mistakes

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Bearings

Bearings

📋 Summary

🧭 What Are Bearings?

The Compass Directions as Bearings

📐 Finding a Bearing Between Two Points

Step-by-Step Method

✏️ Drawing a Point on a Bearing

Step-by-Step Method

Worked Example (from the textbook)

🔄 Back Bearings: Bearing of B from A ↔ A from B

🧩 Trickier Bearings Questions (Pythagoras & Trigonometry)

How These Questions Usually Work

🧠 What to Memorise

✅ Concepts Checklist

🎯 Exam Tips & Common Mistakes

More International Mathematics topics