Scale Factor
Maths: Edexcel IGCSE Higher, topic 4.2.4
Scale Factors
Maps include a scale which tells you the relationship between the distances on the page and the distances in real life.
Distances on maps
- To work out distances from a map, measure the scale given to find out the relationship (e.g. 1cm = 100m).
- Then measure the distance on the map and use the relationship to convert this to the distance in real life.
Scale factors as ratios
- Map scales can be written as a ratio such as 1 : 5000.
- This means that each unit on the map represents 5000 units in real life.
- 1cm on the map = 5000cm in real life.
Ratios in 2D Shapes
2D shapes
- 2D shapes are similar if all pairs of corresponding sides are in the same ratio.
- When a 2D shape is enlarged by a scale factor of S the ratio of each original side to the enlarged side is 1 : S.
Scale Factor and Enlargement
When a shape is enlarged, its lengths are all multiplied by S.
Length
Area
- When a 2D shape is enlarged by a scale factor S, because all the lengths are multiplied by S, its area will be multiplied by S2.
Volume
- When a 3D shape is enlarged by a scale factor S all the lengths are multiplied by S.
- The areas in the shape will be multiplied by S2 and the volumes will be multiplied by S3.
Example
- A cube of side length 1cm is enlarged by scale factor 2.
- Its initial surface area was 6cm2 and its initial volume was 1cm3.
- After enlargement each edge is now 2cm long.
- Its surface area is now 6 × 4cm2 = 24cm2 which is 4 times as large as before.
- Its volume is now 23cm3 = 8cm3 which is 8 times as large as before.
Test yourself on Scale Factor
Which ratio is equivalent to the scale where 1 cm represents 5 km?
Which ratio is equivalent to the scale where 2 cm represents 5 km?
Which ratio is equivalent to the scale where 1 cm represents 3 km?
A map has the scale 1 : 100,000 and the distance between two towns on the map is 5.4 cm. What is the distance between the towns in real life?
A map has the scale 1 cm : 10 km and the distance between two rivers on the map is 3 cm. What is the distance between the rivers in real life?