Unit 19 Section 3 : Line, area and volume scale factors
In this section we look at what happens to the area of shapes and the volume of solids when the lengths
in those shapes or solids are enlarged by a particular scale factor. The examples below will explain further.
Example 1
Consider a rectangle 5cm by 2cm.
The area of the rectangle is 5cm × 2cm = 10cm², but what happens if the rectangle is enlarged?
We will start by multiplying the lengths by scale factor 2.
The rectangle is now 10cm by 4cm and the area is 40 cm².
The lengths were multiplied by 2, but the area has been multiplied by a scale factor of 4.
Now we will multiply the lengths in the original rectangle by scale factor 3.
The rectangle is now 15cm by 6cm and the area is 90 cm².
The lengths were multiplied by 3, but the area has been multiplied by a scale factor of 9.
Finally, we will try multiplying the lengths by scale factor 5.
The rectangle is now 25cm by 10cm and the area is 250 cm².
The lengths were multiplied by 5, but the area has been multiplied by a scale factor of 25.
You should see a pattern! When all the lengths are multiplied by k, the areas are multiplied by k².
Example 2
Consider a cuboid with sides of length 3cm, 4cm and 5cm.
The volume of the cuboid is 3cm × 4cm × 5cm = 60cm³, but what happens if the cuboid is enlarged?
We will start by multiplying the lengths by scale factor 2.
The rectangle is now 6cm by 8cm by 10cm and the volume is 480 cm³.
The lengths were multiplied by 2, but the volume has been multiplied by a scale factor of 8.
Now we will multiply the lengths in the original cuboid by scale factor 3.
The rectangle is now 9cm by 12cm by 15cm and the volume is 1620 cm³.
The lengths were multiplied by 3, but the volume has been multiplied by a scale factor of 27.
Finally, we will try multiplying the lengths by scale factor 10.
The rectangle is now 30cm by 40cm by 50cm and the volume is 60000 cm³.
The lengths were multiplied by 10, but the volume has been multiplied by a scale factor of 1000.
You should see a pattern again – when all the lengths are multiplied by k, the volumes are multiplied by k³.
General Rule
If the lengths in a shape or solid are all multiplied by a scale factor of k, then
the areas will be multiplied by a scale factor of k² and the volumes
will be multiplied by a scale factor of k³.
For example, if the lengths are enlarged with scale factor 4,
then the areas will be enlarged with scale factor 16 and
the volumes will be enlarged with scale factor 64.



Practice Questions
Work out the answer to each of these questions then click on the button marked
to see whether you are correct.
Practice Question 1
A hexagon has area 60 cm².
What will the area of the hexagon be, if it is enlarged with scale factor 3?
Practice Question 2
A cube has all sides of length 2cm.
What are the surface area and volume of the cuboid?
The same cube is enlarged with scale factor 5.
What are the new surface area and volume?
Exercises
Work out the answers to the questions below and fill in the boxes. Click on the
button to find out whether you have answered correctly. If you are right
then will appear and you should move on to the next
question. If appears then your answer is wrong. Click
on to clear your original answer and have another go.
If you can't work out the right answer then click on
to see
the answer.
You have now completed Unit 19 Section 3
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