Bearings, Constructions and Scale Drawings
Geometry
What a scale means
- A scale links a length on the drawing to the matching length in real life, and it is a ratio
- It appears in two forms
- As words, such as "1 cm represents 4 km", which is the form most scale drawings on these papers use
- As a ratio with no units, such as 1 : 25 000, meaning one unit on the drawing is 25 000 of the same units in reality
- A unitless ratio works for any unit, so 1 : 25 000 means 1 cm stands for 25 000 cm, which is 250 m or 0.25 km
- Converting the ratio into "1 cm represents … km" before doing anything else avoids most of the errors here
Working both ways
- To find a real distance, measure the drawing in centimetres and multiply by what one centimetre represents
- To find a drawing length, divide the real distance by what one centimetre represents
- Give the answer in the units the question asks for, converting at the end rather than mid-calculation
- Measure between the centres of the marked points, and to the nearest millimetre
Exam tip
Write the measured length down
When a question asks for a real distance from a scale drawing, write the length you measure, in centimetres, on its own line before using the scale. That measurement earns one of the two marks by itself, and if it is slightly out, multiplying it by the scale still earns the method mark. So for a line of 5.8 cm at 1 cm to 25 km, write 5.8 cm, then 5.8 × 25 = 145 km.
Worked example
Reading a distance off a map
On a map the scale is 1 cm represents 4 km. Two towns are 6.5 cm apart on the map. Find the actual distance between them, and the map distance for a third town 18 km away.
Solution:
- Each centimetre on the map stands for 4 km
- Real distance = 6.5 × 4 = 26 km
- For the second part, divide instead of multiplying
- Map distance = 18 ÷ 4 = 4.5 cm
- Check: 4.5 × 4 = 18
Worked example
Converting a ratio scale
A plan is drawn to a scale of 1 : 25 000. A path measures 8 cm on the plan. Find its real length in kilometres.
Solution:
- The ratio has no units, so 1 cm on the plan stands for 25 000 cm in reality
- Real length = 8 × 25 000 = 200 000 cm
- Convert to metres by dividing by 100: 200 000 ÷ 100 = 2000 m
- Convert to kilometres by dividing by 1000: 2000 ÷ 1000 = 2
- The path is 2 km long