Using length scale factor for area
"The Area Scale Slip"
The Mistake in Action
Two similar shapes have lengths in ratio 1:3. Shape A has area 5 cm². Find the area of shape B.
Wrong: Area of B = 5 × 3 = 15 cm²
Why It Happens
Students know lengths multiply by the scale factor and assume the same applies to area.
The Fix
For similar shapes:
- Length scale factor = $k$
- Area scale factor = $k^2$
- Volume scale factor = $k^3$
Length ratio 1:3 means $k = 3$ Area ratio = $1:3^2 = 1:9$
$$\text{Area of B} = 5 \times 9 = 45 \text{ cm}^2$$
Memory aid: Area is 2D, so square the scale factor. Volume is 3D, so cube it.
Spot the Mistake
Can you identify where this student went wrong?
Lengths in ratio 1:3, Area A = 5 cm²
Area B = 5 × 3 = 15 cm²
Click on the line that contains the error.
Related Topics
Learn more about the underlying maths: