Bearings in GCSE Maths

A bearing is a direction given as an angle. For GCSE Maths there are three rules: measure from north, measure clockwise, and write the angle with three figures, so 50° is written 050°. The bearing of B from A is the angle at A, from the north line clockwise round to the line AB. To go back the other way, add or subtract 180°.

The three rules of bearings

1. Start from north. Draw a north line at the point you are measuring from.
2. Measure clockwise. Turn from north in the direction a clock's hands move.
3. Use three figures. Write 050°, not 50°. Bearings go from 000° to 359°.

The word after "from" tells you where to put the protractor. "The bearing of B from A" means stand at A, face north, and turn clockwise until you face B.

How to measure a bearing

1. Draw a north line at the starting point, if there is not one already.
2. Join the starting point to the other point with a straight line.
3. Put the centre of the protractor on the starting point, with zero along the north line.
4. Measure the angle clockwise. If it is more than 180°, measure the anticlockwise angle and subtract it from 360°.
5. Write the answer with three figures.

AQA GCSE Maths includes bearings with maps and scale drawings (G15), including the eight compass point bearings and three-figure bearings. With a scale drawing, measure the length with a ruler and convert with the scale, for example 1 cm to 2 km.

Back bearings

The bearing of A from B is called the back bearing. North lines at A and B are parallel, so the back bearing is always exactly 180° different:

• If the bearing is less than 180°, add 180°.
• If the bearing is 180° or more, subtract 180°.

Example 1

The bearing of a lighthouse from a boat is 300°. What is the bearing of the boat from the lighthouse? 300° is more than 180°, so subtract: 300° − 180° = 120°.

Example 2: using angle facts

Some questions give the angle between two lines and ask you to work out a bearing with angle rules. Because north lines are parallel, co-interior angles add to 180° and alternate angles are equal. For example, if the bearing of Q from P is 065°, then at Q the angle between the north line and QP, measured anticlockwise, is 180° − 65° = 115° (co-interior angles). The bearing of P from Q is 360° − 115° = 245°, which is 065° + 180°, as expected. More on these rules in parallel lines.

Check your understanding.

Bearings with Pythagoras and trigonometry (Higher)

When a journey makes a right-angled triangle, use SOHCAHTOA and Pythagoras' theorem. For triangles without a right angle, use the sine rule or the cosine rule.

Example 3: finding a bearing

A walker goes 5 km due east, then 8 km due north. Find the bearing of where she ends up from where she started, and how far away it is.

The angle θ is measured from the north line, so the opposite side is the 5 km east and the adjacent side is the 8 km north. The bearing is 032° and the distance is 9.43 km (3 s.f.).

Example 4: splitting a journey

A ship sails 15 km on a bearing of 060°. How far north and how far east has it travelled? The 60° angle is at the start, between north and the ship's path:

Practise finding a missing angle with SOHCAHTOA in this question from the Trigonometry course, or check a triangle in the Trigonometry Calculator.

Common mistakes

• Measuring from the wrong point. "From A" means the protractor goes at A.
• Measuring anticlockwise. Bearings are always clockwise from north.
• Writing two figures. 50° should be 050°.
• Back bearings over 360°. If adding 180° gives more than 360°, you should have subtracted instead.
• Using sin for cos in trigonometry. Label the triangle from the angle at north: the side next to north is adjacent.

Practise with a short timed paper

Bearings come up in GCSE Maths geometry questions on both tiers. The daily GCSE Maths Higher and GCSE Maths Foundation papers are short timed papers of 8 questions, with a new one every day.

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