Bearings and Back Bearings Problem

What Are Bearings and Why Do They Matter?

Bearings are a way of describing direction using angles measured clockwise from north. They are always given as three-figure numbers — so north is 000°, east is 090°, south is 180°, and west is 270°. Bearings appear frequently on the IGCSE Mathematics paper because they combine angle work, trigonometry, and spatial reasoning into a single question.

Many students find bearings questions confusing because they require you to visualise angles from different viewpoints. The concept of a “back bearing” — the bearing you would use to look back at where you came from — is particularly tricky. In this post, we will work through a complete bearings problem to build your confidence.

The Problem

Town A is 40 km from town B on a bearing of 065° from A. Town C is 30 km from town B on a bearing of 150° from B.

(a) Find the bearing of A from B (the back bearing).

(b) Find the angle ABC.

(c) Find the distance from A to C.

(d) Find the bearing of C from A.

Part (a): The Back Bearing

The bearing of B from A is 065°. The back bearing (bearing of A from B) is found using the rule:

If the bearing is less than 180°, add 180°. If the bearing is greater than 180°, subtract 180°.

Since 065° < 180°: Back bearing = 065° + 180° = 245°

This makes geometric sense. If you are facing northeast (065°) from A to reach B, then from B you must face southwest (245°) to look back at A.

Part (b): Finding Angle ABC

To find angle ABC, we need to work out the angle at B between the direction to A and the direction to C.

From B, the bearing of A is 245° (calculated above). From B, the bearing of C is 150° (given).

The angle ABC is the difference between these bearings, but we must be careful about which direction we measure. Visualise standing at B: north is at 000°, the direction to C is at 150° (clockwise from north), and the direction to A is at 245° (clockwise from north).

Angle ABC = 245° − 150° = 95°

This is the angle at B, measured from the direction of C to the direction of A, going clockwise. Since we want the interior angle of triangle ABC, this is correct.

Part (c): Finding Distance AC

We now have a triangle with:

AB = 40 km
BC = 30 km
Angle ABC = 95°

We know two sides and the included angle, so we use the cosine rule:

AC² = AB² + BC² − 2(AB)(BC)cos(ABC)

AC² = 40² + 30² − 2(40)(30)cos(95°)

AC² = 1600 + 900 − 2400 × cos(95°)

Now, cos(95°) = −cos(85°) ≈ −0.0872

AC² = 1600 + 900 − 2400 × (−0.0872)

AC² = 2500 + 209.3

AC² = 2709.3

AC = √2709.3 ≈ 52.1 km

Part (d): Finding the Bearing of C from A

To find the bearing of C from A, we first need to find the angle BAC. We can use the sine rule:

sin(BAC)/BC = sin(ABC)/AC

sin(BAC)/30 = sin(95°)/52.1

sin(BAC) = 30 × sin(95°)/52.1

sin(BAC) = 30 × 0.9962/52.1

sin(BAC) = 29.886/52.1

sin(BAC) = 0.5736

BAC = sin⁻¹(0.5736) ≈ 35.0°

Now, the bearing of B from A is 065°. The bearing of C from A is:

Bearing of C from A = 065° + 35.0° = 100.0°

We add because angle BAC opens clockwise from the direction AB. Always draw a clear diagram to confirm the direction of the addition.

Drawing Clear Diagrams

The single most important strategy for bearings questions is to draw a large, clear diagram. Here is how:

Draw a north line at every point — this is essential. Every town or position needs its own north arrow.
Mark the bearings from north, clockwise — use a protractor if you have one, or estimate the angle.
Label all known distances and angles on the diagram.
Identify the triangle and mark the angle you need.

Without a diagram, bearings questions are almost impossible. With a good diagram, they become manageable.

The Back Bearing Rule Explained

The 180° rule for back bearings works because north lines at different points are parallel. When you draw the north line at A and the north line at B, they are parallel (both point to geographic north). The bearing from A to B and the bearing from B to A are on opposite sides of this parallel arrangement.

If you draw the line AB and the two north lines, the bearing of B from A and the bearing of A from B are co-interior angles (or supplementary, in certain configurations). This is why we add or subtract 180°.

Common Mistakes

Not using three-figure bearings: A bearing of 65° must be written as 065°. Forgetting the leading zero loses a mark.
Measuring anticlockwise instead of clockwise: Bearings are always measured clockwise from north. Measuring anticlockwise gives the wrong answer.
Confusing “bearing of A from B” with “bearing of B from A”: “The bearing of A from B” means “standing at B, looking toward A.” The direction is from B to A.
Incorrectly finding the angle in the triangle: When the back bearing and the given bearing are on the same side of the north line, you subtract. When they cross the north line, you may need to add. Always draw the diagram.
Using the wrong trigonometric rule: Use the cosine rule when you have two sides and the included angle. Use the sine rule when you have a side and the opposite angle.

Practice Questions

The bearing of B from A is 130°. What is the bearing of A from B?
Town P is 50 km from Q on a bearing of 040°. Town R is 35 km from Q on a bearing of 310°. Find the distance PR.
A ship sails 20 km on a bearing of 070°, then 15 km on a bearing of 160°. How far is the ship from its starting point?

Tips for Exam Day

Spend the first 30 seconds drawing a large, clear diagram
Mark every north line before doing any calculations
Write the back bearing rule at the top of your working
Show all cosine rule and sine rule substitutions — method marks are generous
State your final bearing as a three-figure number

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