0580

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 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
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