Town $B$ is $30\text{ km}$ East and $40\text{ km}$ North of Town $A$. Find the three-figure bearing of $B$ from $A$.

1. Ratio : $\tan\theta = \frac{30}{40} = 0.75$. 2. Compute : $\theta = 36.87^\circ$. 3. Three-figure : $\mathbf{037^\circ}$.

Ship $P$ on bearing $050^\circ$ for $12\text{ km}$; ship $Q$ on bearing $130^\circ$ for $15\text{ km}$. Find $PQ$.

1. Included angle : $130 - 50 = 80^\circ$. 2. Cosine rule : $d^2 = 144 + 225 - 360\cos 80^\circ = 306.48\ldots$ 3. State : $d = \mathbf{17.5\text{ km}}$.

Hiker: $5\text{ km}$ on $060^\circ$ from $B$ to $P$, then $8\text{ km}$ on $150^\circ$ from $P$ to $C$. Find the bearing from $C$ back to $B$.

1. Angle at $P$ : back-bearing $240^\circ - 150^\circ = 90^\circ$ (right angle). 2. Angle at $C$ : $\tan C = \frac{5}{8} \implies C = 32.0^\circ$. 3. Back-bearing of leg 2 : $150 + 180 = 330^\circ$. 4. Return bearing : $330 - 32 = \mathbf{298^\circ}$.

A ship sails from a port, $P$, travelling $40 \text{ km}$ on a bearing of $060^{\circ}$ to reach a marker, $M$. At the marker, the ship changes direction and sails $30 \text{ km}$ on a bearing of $150^{\circ}$ to reach an island, $I$. Draw a fully labelled sketch of the ship's journey, showing the North lines at $P$ and $M$. Calculate the exact distance from the port $P$ to the island $I$.

Sketch showing $P$, $M$, $I$ with North lines. Angle at $P$ is $060^{\circ}$, angle at $M$ is $150^{\circ}$. A1 Back bearing of $P$ from $M$ is $060^{\circ} + 180^{\circ} = 240^{\circ}$. M1 Angle $PMI = 240^{\circ} - 150^{\circ} = 90^{\circ}$. The triangle is right-angled at $M$. A1 Using Pythagoras: $PI^2 = 40^2 + 30^2 = 1600 + 900 = 2500$. M1 $PI = 50 \text{ km}$. A1

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL Three-Figure Bearings and Navigation Questions

Updated October 2026

A three-figure bearing is measured clockwise from north and always has three digits (037°, not 37°). IB Maths AI SL bearings questions turn a journey into a triangle: draw a north line at every point, find the angle inside the triangle, then use the sine or cosine rule. Start with the 3 worked examples, then try the exam-style questions and check your working against the mark scheme.

10 questions
Paper 1: 3
Paper 2: 7
2 easy
2 medium
4 hard
2 very hard
3 worked examples

Practise Three-Figure Bearings and Navigation questions →

What you need to know

Three-figure bearings measured clockwise from north. SL AI ships, planes, and hikers — always requiring a right-angled triangle plus the sine or cosine rule. Bearings and navigation problems overview →

What's examined in AI SL three-figure bearings and navigation

The question bank covers these three-figure bearings and navigation question types (number of questions in brackets):

Multi-leg Journeys (7)
Reverse Bearings (2)
Cardinal Point Navigation (1)

Three-Figure Bearings and Navigation worked examples

Worked example 1: Bearing from coordinates · easy

Town $B$ is $30 km$ East and $40 km$ North of Town $A$ . Find the three-figure bearing of $B$ from $A$ .

Solution

3 steps · 1 to fill in

Examiner tip: Bearings are ALWAYS three digits. Leading zeros are mandatory (e.g.

037^{\circ}

Worked example 2: Distance between two ships · medium

Ship $P$ on bearing $050^{\circ}$ for $12 km$ ; ship $Q$ on bearing $130^{\circ}$ for $15 km$ . Find $P Q$ .

Solution

3 steps

Examiner tip: Draw a North arrow at every waypoint to spot included angles.

Worked example 3: Return bearing after two legs · hard

Hiker: $5 km$ on $060^{\circ}$ from $B$ to $P$ , then $8 km$ on $150^{\circ}$ from $P$ to $C$ . Find the bearing from $C$ back to $B$ .

Solution

4 steps · 1 to fill in

Examiner tip: If an internal angle is exactly

90^{\circ}

, use Pythagoras + SOHCAHTOA — faster than sine/cosine rules.

Try these IB Maths AI SL three-figure bearings and navigation questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · medium · 5 marks · Paper 2

A ship sails from a port, $P$ , travelling $40 km$ on a bearing of $060^{\circ}$ to reach a marker, $M$ . At the marker, the ship changes direction and sails $30 km$ on a bearing of $150^{\circ}$ to reach an island, $I$ .

Draw a fully labelled sketch of the ship's journey, showing the North lines at $P$ and $M$ .
Calculate the exact distance from the port $P$ to the island $I$ .

Full worked solution Free, no sign-in

Sketch showing $P$ , $M$ , $I$ with North lines. Angle at $P$ is $060^{\circ}$ , angle at $M$ is $150^{\circ}$ . A1
Back bearing of $P$ from $M$ is $060^{\circ} + 180^{\circ} = 240^{\circ}$ . M1
Angle $P M I = 240^{\circ} - 150^{\circ} = 90^{\circ}$ . The triangle is right-angled at $M$ . A1
Using Pythagoras: $P I^{2} = 40^{2} + 30^{2} = 1600 + 900 = 2500$ . M1
$P I = 50 km$ . A1

Every question here has a full worked solution. See them all →

Question 2 · hard · 6 marks · Paper 2

A boat leaves a port $P$ and sails $20 km$ on a bearing of $040^{\circ}$ to a buoy $B$ . It then turns and sails $35 km$ on a bearing of $140^{\circ}$ to an island $I$ .

Calculate the size of the angle $P B I$ .
Use the cosine rule to calculate the direct distance from the port $P$ to the island $I$ .

Attempt it and see the mark scheme →

Question 3 · hard · 6 marks · Paper 2

Two coastguard vessels, Alpha and Beta, leave the same dock at 12:00. Alpha travels at a speed of $15 km/h$ on a bearing of $050^{\circ}$ . Beta travels at a speed of $22 km/h$ on a bearing of $125^{\circ}$ .

Calculate the distance each vessel has travelled by 14:30.
Calculate the distance between the two vessels at 14:30.

Attempt it and see the mark scheme →

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All 10 three-figure bearings and navigation questions with mark schemes →

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FAQ

How many IB Maths AI SL three-figure bearings and navigation questions are there?

There are 10 exam-style three-figure bearings and navigation questions in the AI SL question bank (Paper 1: 3 · Paper 2: 7), graded 2 easy, 2 medium, 4 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is three-figure bearings and navigation on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 3 · Paper 2: 7. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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IB Maths AI SL Three-Figure Bearings and Navigation Questions

Try it first

What you need to know

What's examined in AI SL three-figure bearings and navigation

Three-Figure Bearings and Navigation worked examples

Worked example 1: Bearing from coordinates · easy

Solution

Worked example 2: Distance between two ships · medium

Solution

Worked example 3: Return bearing after two legs · hard

Solution

Try these IB Maths AI SL three-figure bearings and navigation questions

Question 1 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 2 · hard · 6 marks · Paper 2

Question 3 · hard · 6 marks · Paper 2

Revise three-figure bearings and navigation for the exam

FAQ

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