IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~9 min read
Bearings & Constructions
A bearing is just a clever way to describe a direction — a 3-digit angle measured clockwise from north. Once you can read and draw bearings, every navigation problem becomes a regular trig problem with a fancy disguise.
📘 What you need to know
A bearing is an angle measured clockwise from north, written as a 3-digit number.
Always 3 digits — write 045°, not 45°. Always include leading zeros.
Cardinal directions: N = 000°, E = 090°, S = 180°, W = 270°.
Back bearing from B to A = bearing from A to B ± 180° (add if < 180°, subtract if ≥ 180°).
For most IB problems, draw the diagram, mark the north line(s), then use trig (Pythagoras, SOH CAH TOA, sine/cosine rule) on the resulting triangle.
What is a bearing?
A bearing tells you which direction to head in. Imagine a compass with north at the top. Stand at one point, face north, then turn clockwise until you’re facing your destination. The angle you turned through (in degrees) is the bearing.
⬆
From North
Always start from the north line
↻
Clockwise
Always turn clockwise (never anticlockwise)
3️⃣
3 Digits
Always 3-digit form, e.g. 045° not 45°
A bearing — measured clockwise from north
Read this bearing as “060 degrees” or “060”. The direction shown is roughly east-north-east — 60° clockwise of due north.
The four cardinal directions
The compass has four cardinal points spaced 90° apart. Memorise these — they’re the anchor for all other bearings:
N
North
000°
E
East
090°
S
South
180°
W
West
270°
🧠 Memory trick — “Never Eat Soggy Waffles”
The cardinal directions go N, E, S, W clockwise. So as the bearing increases by 90°, you move clockwise: 000° → 090° → 180° → 270°. Some students remember the order as “Never Eat Soggy Waffles”.
Half-cardinals (NE, SE, SW, NW)
Half way between cardinals you’ve got the diagonals: NE = 045°, SE = 135°, SW = 225°, NW = 315°. They split each 90° quadrant in half.
Back bearings
If you know the bearing from A to B, you can work out the bearing from B back to A — that’s the back bearing. It’s always exactly 180° different.
If you walk in a straight line from A to B, the bearing from B back to A is exactly opposite — a half-turn (180°) different. Adding or subtracting 180° just keeps the answer in the valid 000°–360° range. Think of it as “flipping the arrow around”.
Solving bearings problems with trig
Once you’ve drawn the diagram with bearings marked, the problem usually reduces to a triangle question. The trick is finding the internal angle of the triangle — the angle inside the shape, not the bearing itself.
When a question gives you a bearing, draw a north line at every relevant point and measure the bearing from each. The angles between the bearing lines give you the internal angles of the triangle, which is what you’ll feed into the sine/cosine rule.
The standard 3-step approach
Bearings + trig method
1. Sketch the journey, marking north lines at each point
2. Use the bearings to find the internal angles of the triangle
3. Apply Pythagoras / SOH CAH TOA / sine rule / cosine rule
Typical bearings problem — finding internal angles
Quick angle calculation in the diagram above: the internal angle at B = 180° − 050° + 120° = wait, no — it’s actually (180 − 050) + 120 = 250°… but that’s reflex. The non-reflex internal angle = 360° − 250° = 110°. Always check which side of the angle you need!
When in doubt, mark every angle around point B that you can identify (using parallel north lines and angle facts). Then pick out the one inside your triangle.
Worked examples
WE 1
Express directions as bearings
Write each direction as a bearing in 3-digit form:
(a) due north (b) north-east (c) due south (d) south-west
Step 1: Use the 4 cardinals + half-cardinals(a) North → 000°(b) North-East → halfway between 000° and 090°(b) 045°(c) South → 180°(d) South-West → halfway between 180° and 270°(d) 225°always 3 digits — write 045°, not 45°
WE 2
Find a back bearing
The bearing of a lighthouse from a ship is 075°. Find the bearing of the ship from the lighthouse.
Step 1: Original bearing is 075° < 180°
Use the rule: add 180°.
Step 2: Compute075° + 180° = 255°Bearing of ship from lighthouse = 255°always check: original + back = 360 (or differ by 180)
WE 3
Distance using bearings + Pythagoras
A boat sails 8 km on a bearing of 000° (due north), then turns and sails 6 km on a bearing of 090° (due east). Find the direct distance from its starting point to its current position.
Step 1: Sketch the journey
Two perpendicular legs (north then east) form a right-angled triangle.
Step 2: Apply Pythagorasd2 = 82 + 62 = 64 + 36 = 100Step 3: Square rootd = 10 kmclassic 6-8-10 right triangle (3-4-5 scaled)
WE 4
Two-leg journey using cosine rule
A ship sails 15 km from A to B on a bearing of 040°, then 20 km from B to C on a bearing of 130°. Find the distance AC.
Step 1: Find the internal angle at B
The two bearings differ by 130° − 040° = 90°. But we want the angle inside the triangle. Using the parallel north lines: angle ABC = 180° − 90° = 90°.
turns out it’s a right angle — bonus!Step 2: Apply Pythagoras (or cosine rule with cos 90° = 0)AC2 = 152 + 202 = 225 + 400 = 625Step 3: Square rootAC = 25 kmwhen the bearings differ by 90°, you get a right angle for free
WE 5
Find a bearing from given distances
From a point P, a tower T is 12 km due east, and a lighthouse L is 9 km due north of the tower. Find the bearing of L from P, to the nearest degree.
Step 1: Sketch — right-angled triangle PTL
PT = 12 (east), TL = 9 (north), right angle at T.
Step 2: The angle at P (from north line) is what we want
From P, the north direction points up. Angle between PT (east) and PL (the line we want) is split by the north line.
tan(angle from east) = 912 = 0.75
Angle from east = tan⁻¹(0.75) = 36.87°
Step 3: Convert to bearing (clockwise from north)
Bearing from P = 090° (east) − 36.87° = 053.13°
Bearing of L from P ≈ 053°always state the bearing as a 3-digit number!
💡 Top tips
Always sketch first. A diagram with north lines drawn at every point makes the problem 10× easier.
Draw a north line at every key point. Use a dashed vertical arrow at each landmark — bearings are always measured from these.
Use the 3 rules: from north, clockwise, 3 digits. Get any of these wrong and your answer is wrong.
If the bearing is < 180°, the back bearing is original + 180°. If ≥ 180°, it’s original − 180°. They always differ by exactly 180°.
If two bearings differ by 90° at the same point, the angle between them is a right angle — Pythagoras applies directly.
The angle between two adjacent north lines is the same as a “co-interior” or “alternate” angle — useful for finding triangle angles.
For final bearings: always state your answer as a 3-digit number to the required degree of accuracy.
⚠ Common mistakes
Measuring anticlockwise. Bearings are always clockwise. If you measured the wrong way, your answer is 360° minus the correct value.
Forgetting the leading zeros. Write 045°, not 45°. Examiners deduct marks for missing the 3-digit format.
Using the bearing directly as the internal triangle angle. The internal angle is usually different — you have to work it out using parallel north lines and angle facts.
Confusing back bearing direction. Bearing of B from A is different from bearing of A from B — read the question carefully.
Forgetting to add or subtract 180°. Always check the original is < 180° (add) or ≥ 180° (subtract).
Not drawing a sketch. Bearings problems live and die on the diagram — even a rough sketch helps.
Calculator in radians. Bearings problems are always in degrees — switch your calculator to DEG mode.
Bearings questions are often the most “real-world” feeling problems on the IB paper — a great chance to show off your trig setup skills. Get the diagram right and the rest is just calculation.
Need help with Bearings?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
