Bearings — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers bearings: the way direction is described in navigation, map work and surveying. By the end of this guide you should be able to measure and write a bearing correctly, find the bearing for a return journey, and use bearings together with scale drawings.
You should also be able to combine bearings with angle facts to calculate a direction without measuring, use them alongside trigonometry and the sine and cosine rules, and describe a multi-stage journey.
The organising idea is that every bearing is measured the same way, from a north line at the point you are standing. Three rules follow from that and they are never broken: measure clockwise, measure from north, and write three figures. Almost every mistake in this topic is a failure of one of those three — measuring anticlockwise, measuring from the wrong point, or writing 50° where 050° was needed. Drawing the north line first, at the correct point, prevents nearly all of them.
Key terms and definitions
Bearing — an angle measured clockwise from north, always written with three figures.
North line — the vertical reference line drawn at the point you are measuring from.
Three-figure bearing — a bearing written with three digits, so 50° becomes 050°.
Back bearing — the bearing of the return journey, from the destination back to the start.
Scale drawing — an accurate diagram in which lengths represent real distances.
Due north, east, south, west — the four cardinal directions, at 000°, 090°, 180° and 270°.
Core concepts
The three rules
A bearing is measured clockwise, from north, and written with three figures.
Clockwise means the angle opens in the direction a clock's hands turn, so a direction slightly east of north is a small bearing and a direction slightly west of north is a large one, close to 360°.
From north means from the vertical line pointing up the page, not from any other line in the diagram.
Three figures means 050°, not 50°, and 007°, not 7°. Bearings run from 000° to 360°, and the leading zeros are part of the answer — a bearing written without them loses the mark even when the number is right.
The cardinal directions
North is 000° (or equivalently 360°), east is 090°, south is 180° and west is 270°.
These are worth knowing without thinking, because they give an instant sense check. A bearing of 200° must point roughly south-west; if your diagram shows something heading north-east, the measurement is wrong.
Halfway directions follow: north-east is 045°, south-east is 135°, south-west is 225° and north-west is 315°.
Measuring from the right point
The north line goes at the point you are measuring from, and getting this wrong is the single most common error.
"The bearing of B from A" means stand at A, draw north there, and measure clockwise to the direction of B. "The bearing of A from B" is a different question with a different answer, measured from a north line at B.
Read the wording carefully. The point named after the word "from" is where the north line goes.
Back bearings
The back bearing is the direction of the return journey, and because the two north lines are parallel, the outward and return bearings always differ by exactly 180°.
If the outward bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°.
So a bearing of 070° has a back bearing of 250°, and a bearing of 210° has a back bearing of 030°.
Choosing the operation that keeps the answer between 000° and 360° is the simple way to remember which to use.
Bearings and angle facts
Many questions can be answered by calculation rather than measurement, using the fact that the north lines at two points are parallel.
That makes the familiar parallel-line angle facts available: alternate angles are equal, co-interior angles add to 180°, and angles round a point add to 360°.
For instance, if the bearing of B from A is 070°, then at B the angle between the north line and the line back to A is 70° on the other side, which is exactly why the back bearing works out as 250°.
Angles around a point are used constantly: if a bearing has been found as an angle measured from south or from east, subtract or add as needed to express it from north.
Bearings with scale drawings
Draw a north line at the starting point, use a protractor to measure the bearing clockwise from it, then draw the leg to scale with a ruler.
For a second leg, draw a new north line at the point you have reached, and repeat. The new north line is parallel to the first.
At the end, the distance and bearing back to the start can be measured directly from the drawing and converted using the scale.
Accuracy matters: these questions usually allow a tolerance of about one degree and one or two millimetres, so a sharp pencil and a carefully placed protractor are part of the method.
Bearings with trigonometry
Where the journey forms a right-angled triangle, the direction can be calculated exactly instead of measured.
Find the angle inside the triangle using the appropriate trigonometric ratio, then convert it into a bearing by adding or subtracting from the relevant cardinal direction.
The conversion step is where marks are lost. The angle inside a triangle is rarely the bearing itself — it usually has to be added to 090°, or subtracted from 360°, depending on which quadrant the journey heads into. Sketching the north line at that point makes the conversion obvious.
For triangles without a right angle, the sine and cosine rules do the same job, with the bearings used to work out the angle between two legs of a journey.
Worked examples
Example 1: A back bearing
The bearing of a lighthouse from a ship is 118°. What is the bearing of the ship from the lighthouse?
The outward bearing is less than 180°, so add 180°.
118 + 180 = 298°.
The answer lies between 000° and 360°, which confirms the right operation was chosen. A bearing of 118° points south-east, so the return direction should point north-west, and 298° does.
Example 2: Using angle facts
B is due east of A. C is on a bearing of 150° from A. What is the angle BAC?
Due east from A is a bearing of 090°.
The bearing of C from A is 150°, measured from the same north line.
The angle between the two directions is the difference: 150 − 90 = 60°.
No measuring was needed, because both directions were given as bearings from the same point.
Example 3: Converting a calculated angle into a bearing
A ship sails 8 km due south and then 6 km due west. What is its bearing from the starting point, to the nearest degree?
The two legs form a right angle, so the angle at the start between due south and the final position satisfies tan θ = 6 ÷ 8 = 0.75, giving θ = 36.9°.
That angle is measured from south, not from north, so it must be converted.
Going clockwise from north, south is 180°, and the ship lies a further 36.9° round towards the west: 180 + 36.9 = 216.9°.
To the nearest degree the bearing is 217°.
Quoting 37° as the answer is the error the conversion step exists to prevent.
Common mistakes and how to avoid them
Writing two figures instead of three. A bearing of 50° must be written 050°.
Measuring anticlockwise. Bearings always turn clockwise from north.
Drawing the north line at the wrong point. The north line goes at the point named after the word "from".
Giving an angle inside a triangle as the bearing. Convert it, using the north line at that point.
Adding 180° when you should subtract. Choose whichever keeps the answer between 000° and 360°.
Measuring from a different line in the diagram. Always measure from north, even when another line looks more convenient.
Rounding a scale drawing too casually. These questions carry accuracy marks and usually allow only about one degree of tolerance.
Exam technique for "Bearings"
Draw and label the north line before measuring or calculating anything. It costs a moment and prevents the two commonest errors at once.
Write every bearing with three figures, including leading zeros, from the first line of working onwards.
State which angle facts you are using — "alternate angles, so 70°" — because the reasoning carries marks in calculation questions.
For a multi-stage journey, draw a fresh north line at each turning point.
Sense-check against the cardinal directions: a bearing between 180° and 270° must point somewhere south-west.
In scale-drawing questions, use a sharp pencil, measure to the nearest degree and millimetre, and state the scale you used.
Quick revision summary
Every bearing is measured clockwise, from a north line, and written with three figures — so 50° is written 050°.
The north line is drawn at the point named after the word "from": the bearing of B from A is measured at A.
Cardinal directions: north 000°, east 090°, south 180°, west 270°, with north-east at 045° and south-west at 225°.
A back bearing differs by exactly 180°: add 180° if the bearing is under 180°, subtract it otherwise, choosing whichever keeps the result between 000° and 360°.
North lines at different points are parallel, which makes alternate and co-interior angle facts available for calculating bearings without measuring.
With trigonometry, find the angle inside the triangle first, then convert it into a bearing using the north line at that point — the angle itself is almost never the answer.
For scale drawings, draw a new north line at every turning point and work to about one degree and one millimetre.