Calculating lengths, areas and volumes
Scale factor of enlargement
Proportions can be used to compare lengths of different sized shapes. To work out how much bigger a shape is compared to a smaller one, divide the bigger length by the corresponding smaller length. This is known as the A number which scales or multiples a quantity. of enlargement.
Example
A rectangle has been enlarged. Find the scale factor of enlargement.
To find the scale factor of enlargement, divide one of the larger lengths by the corresponding smaller one.
\(5 \div 2 = 2.5\)
The bigger rectangle is 2.5 times larger than the smaller one. These shapes are called Shapes that have corresponding sides that are proportional and corresponding angles are equal. since all of the lengths have been increased by the same scale factor.
Scale factors of enlargement for area and volume
Scale factors can be used to compare lengths, areas and volumes.
Scale factors are calculated differently for area and volume.
| Length SF | Area SF | Volume SF |
| \(\times 2\) | \(\times 4\) | \(\times 8\) |
| \(\times 3\) | \(\times 9\) | \(\times 27\) |
| \(\times 4\) | \(\times 16\) | \(\times 64\) |
| \(\times n\) | \(\times n^2\) | \(\times n^3\) |
| Length SF | \(\times 2\) |
|---|---|
| Area SF | \(\times 4\) |
| Volume SF | \(\times 8\) |
| Length SF | \(\times 3\) |
|---|---|
| Area SF | \(\times 9\) |
| Volume SF | \(\times 27\) |
| Length SF | \(\times 4\) |
|---|---|
| Area SF | \(\times 16\) |
| Volume SF | \(\times 64\) |
| Length SF | \(\times n\) |
|---|---|
| Area SF | \(\times n^2\) |
| Volume SF | \(\times n^3\) |
This table shows that if a shape’s lengths are increased by a scale factor of 2, the surface area will be increased by a scale factor of 4 and its volume will be increased by a scale factor of 8.
Example
A cuboid is enlarged by doubling all of its lengths.
| Shape 1 | Shape 2 | |
| Lengths | L: 2 cm W: 5 cm D: 2 cm | L: 4 cm W: 10 cm D: 4 cm |
| Surface area | 48 cm2 | 192 cm2 |
| Volume | 20 cm3 | 160 cm3 |
| Lengths | |
|---|---|
| Shape 1 | L: 2 cm W: 5 cm D: 2 cm |
| Shape 2 | L: 4 cm W: 10 cm D: 4 cm |
| Surface area | |
|---|---|
| Shape 1 | 48 cm2 |
| Shape 2 | 192 cm2 |
| Volume | |
|---|---|
| Shape 1 | 20 cm3 |
| Shape 2 | 160 cm3 |
Length scale factor = \(4 \div 2 = 2\)
Surface area scale factor = \(192 \div 48 = 4\)
Volume scale factor = \(160 \div 20 = 8\)