Scale drawings
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Bearings
A bearing describes a direction as an angle measured clockwise from North, always written using three figures.
- Bearings are always measured clockwise from North
- Bearings are always written using exactly three figures, adding a leading zero if needed (e.g. 8° is written as 008°)
- A bearing can be any value from 000° to 360°
- The back bearing (the bearing back to where you came from) differs from the original bearing by exactly 180° - add 180° if the original bearing is less than 180°, or subtract 180° if it is 180° or more
Worked Example: Measuring a Bearing
- Question: Measure the bearing of B from A, given that the angle measured clockwise from North at A to the line AB is 78°.
- Step 1: Bearings are measured clockwise from North
- Step 2: 78° has only two figures, so a leading zero is needed
- Answer: 078°
Worked Example: Finding a Back Bearing
- Question: The bearing of Y from X is 065°. Find the bearing of X from Y.
- Step 1: Since 065° is less than 180°, add 180°
- Step 2: 065+180=245
- Answer: 245°
Common Mistakes
MistakeWriting a bearing with fewer than three figures, e.g. writing 65° instead of 065°
Fixevery bearing must be written using exactly three figures, adding leading zeros as needed
MistakeAlways adding 180° for a back bearing, even when the original bearing is already 180° or more (which would give a value over 360°)
Fixadd 180° if the bearing is less than 180°, but subtract 180° if it is 180° or more, so the answer always stays within 000°-360°
Scale Drawings and Map Scales
A scale drawing represents a real object or area using a smaller (or sometimes larger) version, where every length is in the same fixed ratio to the real length.
- A scale such as "1 cm represents 200 m" means every 1 cm measured on the drawing corresponds to 200 m in real life
- To write a scale in the form 1:n, convert both measurements to the same unit, then simplify so the drawing side of the ratio is 1
- To find a real-life distance, measure the length on the drawing and multiply by the scale factor; to find a length on the drawing, divide the real-life distance by the scale factor
Worked Example: Writing a Scale in the Form 1:n
- Question: A scale drawing uses the scale 1 cm represents 50 m. Write this scale in the form 1:n.
- Step 1: Convert 50 m to cm: 50100=5000 cm
- Step 2: Write as a ratio: 1 : 5000
- Answer: 1:5000
Worked Example: Using a Scale to Find a Real Distance
- Question: On a map with scale 1:25 000, the distance between two towns is measured as 6 cm. Find the real distance in kilometres.
- Step 1: Real distance in cm: 625\,000=150\,000 cm
- Step 2: Convert to km: 150\,0001001000=1.5
- Answer: 1.5 km
Common Mistakes
MistakeForgetting to convert both measurements to the same unit before writing the ratio 1:n
Fixalways convert to a single common unit (usually cm) first, since the ratio itself has no units
MistakeMultiplying by the scale factor when converting from a real distance to a drawing distance, instead of dividing (or vice versa)
Fixgoing from drawing to real life means multiplying by the scale factor; going from real life to the drawing means dividing by it
Interactive revision notes, videos and practice questions load below.