Bearings –

Bearings is a practical geometry topic that appears on both Foundation and Higher tier GCSE Maths papers for AQA, Edexcel, and OCR. A bearing is a way of describing direction using angles measured clockwise from north. You need to know how to measure, draw, and calculate bearings, and at Higher tier you may need to combine bearings with trigonometry or scale drawings. This guide explains the rules, walks you through worked examples at both tiers, covers the most common errors, and provides practice questions. For more angle-based topics, see our angles and parallel lines guide.

What Is a Bearing?

A bearing is an angle that describes the direction of one point from another. Bearings follow three strict rules:

1. They are measured from north.
2. They are measured clockwise.
3. They are written as three figures (e.g. 045°, 120°, 008°).

If the angle is less than 100°, you add a leading zero to make it three digits. For example, an angle of 72° is written as a bearing of 072°.

Key Vocabulary

• Bearing of B from A — stand at A, face north, and turn clockwise until you face B. The angle you turn through is the bearing.
• Back bearing — the bearing of A from B. If the bearing of B from A is θ, the back bearing is θ + 180° (if θ < 180°) or θ − 180° (if θ ≥ 180°).

Important Angle Facts for Bearings

Since all north lines are parallel, when you draw north lines at different points you create parallel lines — and you can use alternate angles, corresponding angles, and co-interior angles between them.

Step-by-Step Method

Measuring a Bearing

1. Draw a north line at the starting point.
2. Place the centre of your protractor on the starting point, aligned with the north line.
3. Measure clockwise from north to the line connecting the two points.
4. Write the bearing as a three-figure number.

Drawing a Bearing

1. Mark the starting point and draw a north line through it.
2. Place your protractor on the point and measure the required angle clockwise from north.
3. Mark the angle and draw a line in that direction.

Calculating a Back Bearing

1. Take the given bearing.
2. If it is less than 180°, add 180°.
3. If it is 180° or more, subtract 180°.

Using Bearings with Trigonometry (Higher)

1. Draw the diagram with north lines at each point.
2. Use the north lines and given bearings to find the angles inside the triangle.
3. Apply the sine rule, cosine rule, or SOHCAHTOA to find missing sides or angles.
4. Convert any calculated angles back into bearings.

Worked Example 1 — Foundation Level

Question: The bearing of town B from town A is 135°. Find the bearing of town A from town B.

Working:

Step 1 — The bearing of B from A is 135°. Since 135° < 180°, the back bearing is: 135° + 180° = 315°

Step 2 — Check: 315° is between 0° and 360° ✓ and is a three-figure number ✓.

Answer: The bearing of A from B is 315°.

Worked Example 2 — Higher Level

Question: A ship sails from port P on a bearing of 060° for 12 km to point Q. It then sails on a bearing of 150° for 9 km to point R. Find the direct distance from P to R and the bearing of R from P. Give your answers to 1 decimal place.

Working:

Step 1 — Find angle PQR.

At Q, the north line and the bearing from P create an angle. The bearing of P from Q (back bearing) = 060° + 180° = 240°. The bearing from Q to R is 150°. The angle PQR (measured inside the triangle) = 240° − 150° = 90°.

Step 2 — Triangle PQR is right-angled at Q. Use Pythagoras: PR² = 12² + 9² = 144 + 81 = 225 PR = 15 km

Step 3 — Find angle QPR using trigonometry: tan(QPR) = 9 / 12 = 0.75 Angle QPR = tan⁻¹(0.75) = 36.9° (1 d.p.)

Step 4 — Bearing of R from P = 060° + 36.9° = 096.9° ≈ 096.9°

Answer: Distance PR = 15 km. Bearing of R from P = 096.9°.

Common Mistakes

• Measuring anticlockwise instead of clockwise. Bearings are always measured clockwise from north. If you measure anticlockwise, subtract your angle from 360°.
• Forgetting the three-figure rule. A bearing of 45° must be written as 045°. Dropping the leading zero loses marks.
• Drawing the north line in the wrong place. The north line must be drawn at the point you are measuring from, not at the destination.
• Getting the back bearing wrong. Remember: add 180° if the bearing is under 180°; subtract 180° if it is 180° or over. A common error is always adding.
• Not drawing parallel north lines. When working with multiple points, draw a north line at every point. These parallel lines help you find angles using alternate and co-interior angle rules.

Exam Tips

• Always draw a clear diagram with north lines at every relevant point. Even a rough sketch helps you identify the angles.
• Mark angles carefully. Use arcs to show which angle you mean — bearings questions often have several angles in the same diagram.
• For Higher tier, be ready to use the sine rule or cosine rule in non-right-angled bearing triangles.
• Use alternate angles between parallel north lines — this is the fastest way to find interior angles of the triangle.
• Read the question carefully. "Bearing of B from A" means you stand at A and look towards B. Getting this the wrong way around changes the answer completely.

Practice Questions

Question 1: Write the bearing for an angle of 7° from north.

Question 2: The bearing of X from Y is 210°. Find the bearing of Y from X.

Question 3: The bearing of B from A is 075°. The bearing of C from A is 140°. Find angle BAC.

Question 4: A walker goes from A on a bearing of 090° for 5 km to B, then on a bearing of 180° for 8 km to C. Find the direct distance AC.

Question 5: From the previous question, find the bearing of C from A to the nearest degree.

Sharpen your bearing skills with personalised practice at GCSEMathsAI — our AI adapts to your level and provides step-by-step solutions.

Related Topics

• Angles: Basic Rules and Parallel Lines — essential angle facts used in bearing problems.
• Trigonometry — sine and cosine rules for Higher bearing questions.
• Constructions and Loci — accurate drawing with compasses and protractors.
• Vectors — describing direction and magnitude.

Summary

Bearings describe direction using angles measured clockwise from north, always written as three-figure numbers. The back bearing is found by adding or subtracting 180°. At Foundation level, you need to measure, draw, and calculate simple bearings. At Higher level, you combine bearings with trigonometry to find distances and angles in more complex scenarios. The keys to success are drawing clear diagrams with north lines at every point, using parallel line angle rules, and remembering the three strict bearing rules: from north, clockwise, three figures. Practise reading and drawing bearings until the process is automatic.