Bearings and Geometric Drawings
A bearing is a direction measured clockwise from north at the starting point. The biggest source of error is using the wrong origin, so every solution begins by deciding exactly where the bearing is measured from.
Foundation & Higher. Basic scale work is already covered in Units, Scale Drawings and Maps. This lesson applies that skill to geometry rather than reteaching it.
Three-figure bearings
Bearings are measured clockwise from north and written with three figures, for example 045°, 120° or 305°. A bearing of 45° must be written 045° in exam notation.
The eight compass directions are N 000°, NE 045°, E 090°, SE 135°, S 180°, SW 225°, W 270° and NW 315°. North may also be written 360° when completing a full turn.
‘The bearing of A from B’ means stand at B, face north at B, then turn clockwise until you face A.
Reverse bearings
The reverse bearing differs by 180°. If the original bearing is less than 180°, add 180°. If it is 180° or more, subtract 180°.
Example: A from B is 072°, so B from A is 072°+180°=252°.
Parallel north lines and internal angles
North lines at different points are parallel. That lets you use corresponding, alternate and co-interior angle facts when a route crosses them.
Do not assume the difference between two bearings is automatically the internal angle of a triangle. Sketch the north lines and identify the actual angle required.
Measuring and drawing a bearing
1. Mark north at the starting point. 2. Put the protractor centre on that point. 3. Measure clockwise from north. 4. Draw the direction ray. 5. If a distance is given, apply the stated scale along the ray.
For a construction-style question, accuracy of the north line, angle and scaled distance all matter.
Worked example — intersecting bearings
A lighthouse L is on a bearing of 060° from harbour H. A buoy B is on a bearing of 130° from H. The angle LHB is therefore 130°−60°=70° because both bearings are measured from the same north line at H.
If the bearings came from different starting points, you would first draw parallel north lines and work out the internal angles carefully.
Scale-route drawing
Suppose a walker travels 6 km on a bearing of 040°, then 4 km on a bearing of 120°. With scale 1 cm : 1 km, draw 6 cm at 040° from the start, then draw a fresh north line at the new point and 4 cm at 120°.
This produces a geometric drawing that can later support measurements or trigonometry.
Think Like a Mathematician
A bearing is not an angle floating in space. It is an angle tied to an origin and a north line. If the origin changes, redraw north there before doing anything else.
Common Mistakes
Measuring anticlockwise. Bearings are clockwise from north.
Using the destination as the origin. Read ‘A from B’ carefully.
Writing 45° instead of 045°. Bearings use three figures.
Subtracting two bearings from different origins without a diagram.
Practice — bearings and drawings
1. Write a bearing of 7° using three figures.
007°.
2. A direction is due east. State its bearing.
090°.
3. A direction is due south. State its bearing.
180°.
4. A from B has bearing 064°. Find B from A.
064+180=244°.
5. A from B has bearing 235°. Find B from A.
235−180=055°.
6. Two rays from the same point have bearings 035° and 110°. Find the smaller angle between them.
110−35=75°.
7. Why must north be drawn at the starting point of a bearing?
Because the bearing is measured clockwise from north at that origin.
8. A map uses 1 cm : 2 km. Draw a route 14 km long on bearing 125°. How long should the drawn route be?
14÷2=7 cm. Draw north at the start, measure 125° clockwise with a protractor, and draw a 7 cm segment in that direction.
9. A student measures bearing A from B at A. Explain the error.
‘A from B’ means the starting point is B, so the north line and angle must be at B.
10. The bearing of C from D is 300°. Is the reverse bearing 120° or 480°?
Subtract 180°: 120°. Use 000°≤bearing<360°; 360° is an alternative notation for north.
11. From point P, Q is on bearing 025° and R is on bearing 155°. Find angle QPR.
Both bearings share origin P, so 155−25=130°.
12. Why can the difference between bearings from different starting points be misleading?
Each bearing is measured from a different north line. Draw the parallel north lines and identify the triangle angle before calculating.
Frequently Asked Questions
Do bearings always need three digits?
Yes in standard GCSE notation: 7° is 007° and 45° is 045°.
Can I use reverse bearing by adding 180° every time?
Add 180° only when that keeps the result below 360°. Otherwise subtract 180°.
Is trigonometry needed for bearings?
Not for this lesson. Right-triangle and non-right-triangle trig applications come later.