Three-figure bearings and return directions
Measure clockwise from north at the correct starting point and find reverse bearings.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Measure an angle with a protractor.
- Use a scale drawing.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the bearing of due east?
What you will learn
- Read and write bearings using three digits.
- Find a return bearing by changing direction by 180°.
- Combine a bearing with a scale distance to draw a route.
Start at north at the starting point
The bearing of B from A is measured at A. Draw a north line through A, then measure clockwise from that north line to AB. North is 000° (also a full turn of 360°), east 090°, south 180° and west 270°. Write 35° as 035° to preserve the three-figure convention.
A bearing of 145° points southeast: it is 55° short of south. A diagram's small angle to south is therefore not the bearing. The word from identifies the starting point, even when the named destination comes first. Always label your north line and the clockwise angle on paper.
Reverse the journey
The return bearing changes direction by 180°. If the outward bearing is less than 180°, add 180°; if it is at least 180°, subtract 180°. For A to B at 145°, B to A is 325°. Parallel north lines at A and B give the same reference direction, so the reverse direction is exactly half a turn away.
To draw a journey of 4 km on bearing 060° at 1 cm per km, mark 60° clockwise from north and draw a 4 cm segment along that ray. For a second leg, draw a new north line at the new starting point before measuring its bearing. Do not measure the second bearing relative to the previous route unless the question explicitly gives a turning angle.
Pause and explain
A to B has bearing 248°. What is B to A?
Worked example
The bearing of B from A is 145°. Find the bearing of A from B.
Show the worked solution
- The starting point for the requested bearing is now B.
- The direction back to A is 180° different: 145 + 180 = 325.
- Write the angle as the three-figure bearing 325°.
Answer 325°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the reverse bearing of 072°.
Give me a hint
The given bearing is less than 180°.
Compare my reasoning
- Reversing changes direction by 180°.
- 72 + 180 = 252 degrees.
- Write the result as 252°, clockwise from north at the return starting point.
252°.
Look for these in your work
- Changed the starting point.
- Used a half-turn rather than a reflection.
Find the reverse bearings of 310° and 180°.
Give me a hint
Subtract 180° when the bearing is at least 180°.
Compare my reasoning
- 310 − 180 = 130 degrees.
- 180 − 180 = 0 degrees, which is north.
- Write the three-figure answers 130° and 000°.
130° and 000°.
Look for these in your work
- Kept both bearings within one turn.
- Included leading zeroes where needed.
Draw a 3 km leg from P on bearing 060°, then a 2 km leg from Q due east, using 1 cm per km. Name the return bearing of the second leg.
Give me a hint
Draw a fresh north line at Q before the second leg.
Compare my reasoning
- From P draw a 3 cm segment 60° clockwise from north to Q.
- From Q draw a 2 cm segment on 090° to R.
- Returning from R to Q is due west, bearing 270°. Check both lengths with a ruler.
Paper route with 3 cm and 2 cm legs; R to Q is 270°.
Look for these in your work
- Measured each bearing at its own starting point.
- Used the specified scale and three-figure return bearing.
Common mistakes
- Measuring anticlockwise from north.
- Using south or the previous leg as the reference.
- Writing 68° instead of the three-figure 068°.
What can you explain now?
The bearing of Q from P is 025°. At which point is the angle measured?
Compare with the explanation
P
The word from names the starting point, whose north line is the bearing reference.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch two directions with bearings below and above 180°. Add north lines at both endpoints and label the return bearings.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.