Bearings and Scale
Section: Position and Transformation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is a Bearing?
A bearing gives a direction as an angle measured clockwise from North, always written using three figures (e.g. 007°, 090°, 245°).
A bearing of 065°, measured clockwise from North
| Direction | Bearing |
|---|---|
| North | 000° |
| East | 090° |
| South | 180° |
| West | 270° |
Back Bearings
The back bearing is the bearing back from the second point to the first. Add 180° if the original bearing is less than 180°; subtract 180° if it is 180° or more.
-
Question: The bearing of B from A is 065°. Find the bearing of A from B.
- Step 1: 065° is less than 180°, so add 180°: 065 + 180
- Answer: 245°
-
Question: The bearing of B from A is 230°. Find the bearing of A from B.
- Step 1: 230° is 180° or more, so subtract 180°: 230 − 180
- Answer: 050°
Scale on Maps and Plans
A scale like 1:50000 means 1 unit on the map equals 50000 of the same units in real life. Multiply a map distance by the scale factor to find the real distance; divide to go the other way.
-
Question: A map has scale 1:25000. Two towns are 6 cm apart on the map. Find the real distance.
- Step 1: Real distance = 6 × 25000 = 150,000 cm
- Step 2: Convert to a sensible unit: 150,000 cm = 1500 m = 1.5 km
- Answer: 1.5 km
-
Question: Two cities are 20 km apart in real life. On a map with scale 1:100000, how far apart are they on the map?
- Step 1: Convert 20 km to cm: 20 × 100,000 = 2,000,000 cm
- Step 2: Divide by the scale factor: 2,000,000 ÷ 100,000
- Answer: 20 cm
Combining Bearings and Scale
Bearings and scale are often used together to interpret a route or position on a map.
-
Question: A boat travels 8 cm on a map from a harbour to a buoy, on a bearing of 040°. The map's scale is 1:20000. Find (a) the real distance travelled, and (b) the bearing of the harbour from the buoy.
- Step 1 (real distance): 8 × 20000 = 160,000 cm = 1600 m = 1.6 km
- Step 2 (back bearing): 040° is less than 180°, so add 180°: 040 + 180 = 220°
- Answer: (a) 1.6 km, (b) 220°
Locating a Point Using Two Bearings
A single bearing only tells you a DIRECTION from one point - the exact position isn't fixed until you know where the point lies along that direction. If you're given bearings from TWO different reference points instead, the target point must lie on both rays at once, so it's fixed at the point where the two rays cross.
Point C is fixed only where the two bearing rays from X and Y cross
-
Question: A campsite, C, is on a bearing of 050° from a car park, X, and on a bearing of 320° from a visitor centre, Y. Describe how to find the position of C on a diagram showing X and Y.
- Step 1: At X, draw a North line, then measure 050° clockwise from North with a protractor and draw a ray in that direction - C lies somewhere on this ray
- Step 2: At Y, draw a North line, then measure 320° clockwise from North and draw a ray in that direction - C lies somewhere on this ray too
- Step 3: Mark C at the point where the two rays cross
- Answer: C is located at the intersection of the two bearing rays from X and Y
Real-World Applications
Bearings and scale are used throughout navigation and mapping:
- Navigation: ships and aircraft use bearings to hold a precise direction.
- Orienteering and hiking: reading a compass bearing against a map.
- Surveying: locating boundaries and features precisely.
- Search and rescue: directing search parties along exact bearings.
- Road maps and floor plans: reading real-world distances from a scale.
Exam Tips
Common Mistakes
MistakeWriting a bearing with fewer than three figures, e.g. "65°" instead of "065°"
Fixbearings are always written using three figures, e.g. 065°, 007°, 340°
MistakeMeasuring a bearing anticlockwise from North
Fixbearings are always measured CLOCKWISE from North
MistakeAlways adding 180° for a back bearing, even when the original bearing is already 180° or more
Fixadd 180° if the bearing is under 180°; subtract 180° if it is 180° or more
MistakeLeaving a scale answer in cm instead of converting to a sensible unit like km
Fixalways check what unit the question wants, and convert your final answer
MistakeDividing instead of multiplying when converting a map distance to a real distance
Fixmultiply the map distance by the scale factor to get the real distance; divide to go the other way
MistakeDrawing only one bearing ray and guessing where the target point is
Fixone bearing only gives a direction, not a fixed position - you need TWO bearings from two different points, and the answer is where the rays cross
For Exams
- Always use three figures for a bearing, measured clockwise from North.
- Learn the key bearings: North = 000°, East = 090°, South = 180°, West = 270°.
- Back bearings: add 180° (if under 180°) or subtract 180° (if 180° or more).
- Convert your final answer to sensible real-world units (e.g. km, not cm).
- Sketch a quick North arrow to check your bearing makes sense.
Interactive revision notes, videos and practice questions load below.