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 Click on this button below to see the correct answer 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 Click this button to see if you are correct 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 Click on this button to see the correct answer to see the answer.

Question 1
Two similar rectangles are shown below:
(a) Calculate the area of rectangle A.
cm²

(b) Calculate the area of rectangle B.
cm²

(c) What scale factor produces the enlargement from A to B?

(d) How many times bigger than the area of A is the area of B?

Question 2
A rectangle is shown below:
What would the new area of the rectangle be if it was enlarged with:
(a) scale factor 2        cm²
(b) scale factor 3        cm²
(c) scale factor 6        cm²
(d) scale factor 10      cm²
Question 3
The table below gives information about enlargements of the triangle shown, which has an area of 6cm².
Fill in the missing values in the table.
Length
of base
Height of
triangle
Scale
factor
New
area
Area scale
factor
Check your
answers
3cm
4cm
1
6cm²
1
cm
cm
2
cm²
cm
12cm
cm²
cm
16cm
cm²
15cm
cm
cm²
cm
cm
6
cm²
30cm
40cm
600cm²
100
4.5cm
cm
cm²

Question 4
The parallelogram shown has an area of 42cm². It is to be enlarged with a scale factor of 5.
What will the area of the enlarged parallelogram be?
cm²
Question 5
The area of a circle is 50cm² . A second circle has a radius that is 3 times the radius of the first circle.

What is the area of the second circle?
cm²

Question 6
Look at the two cuboids below:
(a) What scale factor is needed to enlarge the smaller cuboid to become the larger?

(b) How many of the smaller cuboids can be fitted into the larger cuboid?

(c) What must the volume of the smaller cuboid be multiplied by to get the volume of the larger cuboid?

Question 7
The table below gives information about enlargements of the cuboid shown, which has a volume of 36cm³.
Fill in the missing values in the table.
Width
Length
Height
Scale
factor
New
volume
Volume
scale factor
Check your
answers
3cm
6cm
2cm
1
36cm³
1
6cm
cm
cm
2
cm³
cm
cm
cm
4
cm³
cm
cm
10cm
cm³
30cm
cm
cm
cm³

Question 8
A tank has a volume of 32m³. It is enlarged with scale factor 3.

What is the volume of the enlarged tank?


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Produced by A.J.Reynolds August 2008