Congruency, Enlargement and Combined Transformations · 5.2.4
Area ratio in enlargement
This standard asks students to investigate, through examples, how enlarging a shape by scale factor k changes its area, discover that the image's area becomes k² times the object's area, verify this relationship with calculations, and then apply it confidently.
The official learning standard (5.2.4)
“Make and verify conjecture on the relation between area of the image and area of the object of an enlargement.”
What it means
This standard asks students to investigate, through examples, how enlarging a shape by scale factor k changes its area, discover that the image's area becomes k² times the object's area, verify this relationship with calculations, and then apply it confidently.
How it is examined
Paper 1 questions often give the scale factor and object area directly, asking students to calculate the image area using k². Paper 2 questions may require finding the scale factor from given areas first, then applying it within a larger enlargement or geometry problem.
Worked example
A triangle has an area of 5 cm². It is enlarged with a scale factor of 4.
Find the area of the image.
- State the relationship between the areas: Area of image = k² × Area of object, where k is the scale factor.
- Substitute the given values: k = 4 and Area of object = 5 cm².
- Calculate k²: 4² = 16.
- Multiply: Area of image = 16 × 5 = 80 cm².
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why is the area scaled by k² and not k?
Because area depends on two dimensions (length × width), so when every length is multiplied by k, the area is multiplied by k × k = k². This can be checked by enlarging a simple square and comparing the two areas.
How do I find the scale factor if I only know the areas?
Divide the image's area by the object's area to get k², then take the square root: k = √(Area of image ÷ Area of object).
Does this k² rule apply to all shapes, not just squares or triangles?
Yes, it applies to any two-dimensional shape, circles, polygons, or irregular regions, because enlargement scales every linear dimension by k equally, so area always scales by k² regardless of the shape's form.