Geometry and Measurement · Measurement · Lesson 7
Back Bearings: Finding the Way Home
If you leave Accra for Cape Coast on one bearing, what bearing must you follow to return? The answer is the back bearing. Ship captains, hunters and hikers use it to retrace a route without getting lost. You will learn a very short rule and understand exactly why it works.
By the end you will be able to
- ✔Define a back bearing as the bearing of the starting point from the finishing point.
- ✔Find the back bearing when the given bearing is less than 180° by adding 180°.
- ✔Find the back bearing when the given bearing is greater than 180° by subtracting 180°.
- ✔Explain why the rule works using parallel North lines and a half turn.
Key words
- Back bearing
- The bearing of the starting point (P) as seen from the finishing point (Q); the direction you face to go straight back.
- Forward bearing
- The bearing of the place you are travelling to, measured at the starting point.
- Parallel lines
- Straight lines that never meet; all North lines point the same way, so they are parallel.
- Half turn
- A turn of 180°, which makes you face exactly the opposite direction.
Curriculum reference
Content standard: B7.3.2.3 Demonstrate understanding of bearings, vector and its components using real life cases.
Indicator: Explain how to find the back bearing when the direction of travel has a bearing which is less than 180° and/or greater than 180°.
Starter
Lost on the way back?
Kwame and Abena hike from a camp in the Aburi hills to a waterfall on a bearing of 060°. On the way back, the mist comes down and they can't see the camp. Kwame says 'Just reverse the numbers: 060 becomes 006!' Abena shakes her head: 'No, I have to turn right round.' Who is right? By the end of this lesson you will be able to settle the argument with a rule that never fails.
1. The forward bearing and the back bearing
Suppose a fisherman sails from harbour P to an island Q. The bearing of Q from P is the forward bearing. To return, he must work out the bearing of P from Q. This is the back bearing.
Think about facing a direction and then turning to face exactly the opposite way. You make a half turn, which is 180°. So the back bearing differs from the forward bearing by 180°. The only question is whether we add 180° or subtract it. The reason is that a bearing must always lie between 000° and 360°.
▶ Follow the four steps: the forward bearing at P, the parallel North line at Q, and the half turn that gives the back bearing.
2. The rule: add 180° or subtract 180°
Because a bearing cannot be more than 360°, we use two cases.
- If the forward bearing is less than 180°, add 180°. Example: 060° + 180° = 240°.
- If the forward bearing is greater than 180°, subtract 180°. Example: 250° − 180° = 070°.
| Forward bearing | Size | Calculation | Back bearing |
|---|---|---|---|
| 040° | less than 180° | 040° + 180° | 220° |
| 105° | less than 180° | 105° + 180° | 285° |
| 210° | more than 180° | 210° − 180° | 030° |
| 300° | more than 180° | 300° − 180° | 120° |
Why does this work? Look at the diagram in the animation. The North lines at P and at Q are parallel. The line PQ crosses them, so the angle at P (the bearing, 060°) and the angle between the South-pointing part of the North line at Q and QP are alternate angles, and alternate angles are equal. Walking round from North at Q clockwise, we first turn 180° to face South and then another 060°, giving 180° + 060° = 240°.
3. Working with 'P from Q' statements
The curriculum often phrases the question like this: 'The bearing of P from Q is 060°. What is the bearing of Q from P?' Read slowly. The bearing is given from Q, so Q is where you stand first. The question asks you to swap the roles: stand at P and look at Q.
- 1Write down the given bearing.
- 2Decide: is it less than 180° or greater than 180°?
- 3Add or subtract 180° accordingly.
- 4Write the answer in three figures and check it lies between 000° and 360°.
- 5Check sense: the new direction should be roughly opposite.
Using the rule: the bearing of P from Q is 060°, so the bearing of Q from P is 060° + 180° = 240°. If the bearing of P from Q is 145°, the bearing of Q from P is 145° + 180° = 325°.
4. Back bearings in journeys and equations
Back bearings appear in longer problems. If Yaw cycles from Ho to Hohoe on a bearing of 025°, then the bearing of Ho from Hohoe is 025° + 180° = 205°. To return to Ho he cycles on 205°, covering the same distance.
Sometimes the bearings are algebraic. If the bearing of A from B is 245° and the bearing of B from A is (x + 15)°, we know the two bearings are 180° apart, so (x + 15) = 245 − 180 = 65, giving x = 50.
| Journey | Forward bearing | Return bearing |
|---|---|---|
| Accra → Tema (illustration) | 070° | 250° |
| Sunyani → Techiman (illustration) | 010° | 190° |
| Cape Coast → Takoradi (illustration) | 255° | 075° |
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Bearing less than 180°
▶ The forward bearing 060° plus a half turn gives the back bearing 240°.
Show the solution
- Step 1060° is less than 180°, so add 180°.
- Step 2060° + 180° = 240°.
- Step 3Check: Q is North-East of P, so P is South-West of Q. 240° is in the South-West area. ✓
Answer: 240°
Example 2
Bearing between 90° and 180°
Show the solution
- Step 1145° is less than 180°, so add 180°.
- Step 2145° + 180° = 325°.
- Step 3325° is between 270° and 360° (North-West): Q is North-West of P while P is South-East of Q. ✓
Answer: 325°
Example 3
Bearing greater than 180°
Show the solution
- Step 1250° is greater than 180°, so subtract 180°.
- Step 2250° − 180° = 070°.
- Step 3Write in three figures: 070°.
Answer: 070°
Example 4
Close to North
Show the solution
- Step 1315° > 180°, so subtract 180°.
- Step 2315° − 180° = 135°.
- Step 3Check: North-West and South-East are opposite. ✓
Answer: 135°
Example 5
A bearing in an equation
Show the solution
- Step 1The two bearings differ by 180°.
- Step 2245° is greater than 180°, so the bearing of B from A is 245° − 180° = 65°.
- Step 3x + 15 = 65, so x = 65 − 15 = 50.
- Step 4Check: 50 + 15 = 65; 65 + 180 = 245 ✓
Answer: x = 50
Summary
- A back bearing is the bearing of the starting point seen from the end point.
- Forward and back bearings are exactly 180° apart.
- If the bearing is less than 180°, add 180°; if it is greater than 180°, subtract 180°.
- A bearing never exceeds 360° and is always written with three figures.
- The distance on the return journey stays the same.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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