Formula sheet
Area of image
area of image = k² × area of object is given on the exam formula sheet. You must still remember to square the scale factor (because area is two-dimensional), and to use the same length units throughout.
What the symbols mean
- k Scale factor of the enlargement (no unit).
- area of object Area of the original shape before enlargement (area unit).
- area of image Area of the enlarged shape (area unit).
Given in the exam, or memorise?
area of image = k² × area of object is given on the exam formula sheet. You must still remember to square the scale factor (because area is two-dimensional), and to use the same length units throughout.
Why it works
Enlarging every length by k multiplies area by k².
- Under scale factor k, every length is multiplied by k.
- Area depends on two lengths (like base × height), each scaled by k.
- So area is multiplied by k × k = k²: area of image = k² × area of object.
Worked example 1
A shape of area 12 cm² is enlarged by scale factor 3. Find the area of the image.
- Area of image = k² × area of object
- = 3² × 12
- = 9 × 12
- = 108
Worked example 2
An object of area 4 cm² is enlarged to an image of area 100 cm². Find the scale factor.
- Area of image = k² × area of object
- 100 = k² × 4
- k² = 100 ÷ 4 = 25
- k = √25 = 5
Where students go wrong
- Multiplying the area by k instead of k², areas scale by the square of the length factor.
- Forgetting to take the square root when solving for k from areas.
- Using k² as if it were the ratio of lengths rather than the ratio of areas.
- Mixing area units, e.g. combining cm² and mm² without converting.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Why is it k² and not just k for area?
Because area is measured in two dimensions. When every length grows by k, both the width and the height grow by k, so the area grows by k × k = k².
For example a scale factor of 3 makes lengths three times longer but area nine times larger. Lengths use k, areas use k².
How do I find the scale factor from two areas?
Divide the image area by the object area to get k², then take the square root to get k. For instance 100 cm² over 4 cm² gives k² = 25, so k = 5.
The square root step is essential: the ratio of areas is k², not k, so you must undo the square to reach the length scale factor.
Does the same idea apply to volume?
Yes, but volume uses k³ because it is three-dimensional. Lengths scale by k, areas by k², and volumes by k³.
So a scale factor of 2 doubles lengths, quadruples areas, and multiplies volumes by eight. Match the power to the number of dimensions: 1 for length, 2 for area, 3 for volume.