Less common M3.1

Scale Drawings and Maps

This topic covers how to relate measurements between mathematically similar shapes, diagrams, and maps using a consistent multiplier called a scale factor. It's a key skill for interpreting scaled representations and converting between map distances and real-world dimensions without a calculator.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Similar shapes have identical corresponding angles and their corresponding side lengths are in the same ratio.
  • The scale factor (k) is the number you multiply an original length by to get the length of the corresponding side on the new shape.
  • A scale factor greater than 1 represents an enlargement, while a scale factor between 0 and 1 represents a reduction.
  • Map scales are often given as a ratio, e.g., 1:50,000. This means 1 unit of length on the map represents 50,000 of the same units in reality.
  • Area and volume do not scale by the length scale factor. If length scales by k, area scales by k2 and volume scales by k3.

Formulae

New Length = k × Original Length

To find a corresponding length on a similar shape, where 'k' is the linear scale factor.

New Area = k2 × Original Area

To find the area of a similar shape when the original area and linear scale factor 'k' are known.

New Volume = k3 × Original Volume

To find the volume of a similar 3D shape when the original volume and linear scale factor 'k' are known.

Definitions

Similar Shapes
Two shapes where one is an enlargement of the other. All corresponding angles are equal and the ratio of corresponding side lengths is constant.
Scale Factor
The constant ratio by which all lengths of a shape are multiplied to produce a similar shape. It is calculated as New Length / Original Length.
Map Scale
The ratio that connects a distance on a map to the corresponding distance on the ground. It is typically unitless, for example, 1:100,000.

Worked example

A map is drawn to a scale of 1:500,000. On this map, a rectangular nature reserve is 2 cm long and 0.4 cm wide. What is the actual area of the nature reserve in square kilometres (km2)?

  1. 1

    First, interpret the scale.

    1:500,000 means 1 cm on the map represents 500,000 cm in reality.

  2. 2

    Convert the real-life distance into a more useful unit.

    Since 100,000 cm = 1 km, then 500,000 cm = 5 km.

    So, the scale is 1 cm :

    5 km.

  3. 3

    Calculate the actual dimensions of the nature reserve using this converted scale.

  4. 4

    Actual Length = Map Length × 5 km/cm = 2 cm × 5 km/cm = 10 km.

  5. 5

    Actual Width = Map Width × 5 km/cm = 0.4 cm × 5 km/cm = 2 km.

  6. 6

    Calculate the actual area.

    Area = Actual Length × Actual Width = 10 km × 2 km = 20 km2

Answer: 20 km2

Common mistakes

  • ×Making unit conversion mistakes with map scales, for example forgetting that 1 km = 100,000 cm.
  • ×Applying the length scale factor directly to areas. Remember to square the length scale factor (k) to find the area scale factor (k2).
  • ×Confusing the direction of the scale factor. To get from a small shape to a large one, k > 1. To get from a large shape to a small one, k < 1.

No-calculator tips

  • ✓Before performing calculations, convert the map ratio (e.g., 1:250,000) into a direct relationship between units, like '1 cm represents 2.5 km'. This avoids manipulating very large numbers.
  • ✓When squaring conversion factors, be careful with zeros. For example, since 1 m = 100 cm, 1 m2 = (100)2 cm2 = 10,000 cm2, not 200 cm2.
  • ✓Work with fractions where possible. A scale factor of 0.25 is easier to handle as 1/4. Multiplying by 1/4 is the same as dividing by 4.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 1 topics