🎯 Learning Objectives
By the end of this lesson, you should be able to:
- Explain why area changes by the square of the scale factor.
- Calculate area factor from linear scale factor.
- Find an image area or original area.
- Avoid using the linear factor directly on area.
🤔 Start with the Idea
When every length is multiplied by a scale factor, both the length and width of an area change. That is why area does not change by the same factor as length.
📖 From Length to Area
area factor = (linear scale factor)²
If k = 3, then every length triples. A 1 by 1 square becomes 3 by 3, so its area becomes 9 times as large.
3² = 9
For areas, square the linear scale factor. For volumes, the corresponding rule is to cube it.
🎮 Try It Yourself
Change the corresponding lengths and watch the scale factor update.
🧠 Let’s Think Together
Two similar shapes have linear scale factor 2.5 from the smaller to the larger. The smaller area is 32 cm².
Area factor = 2.5² = 6.25.
larger area = 32 × 6.25 = 200 cm²
Answer: 200 cm².
✏️ Practice Questions
Build confidence- Find the area factor when k = 4.
- A shape of area 18 cm² is enlarged with k = 3. Find the image area.
- Two similar regions have area ratio 49 : 81. Find the linear scale factor from first to second.
✅ Check your answers
- 16
- 162 cm²
- 9/7
⚠️ Common Mistakes
- Using different units on the two sides of a ratio scale.
- Multiplying when the question requires division, or dividing when it requires multiplication.
- Leaving out the unit or using a linear unit for an area.
- Rounding too early in a multi-step calculation.
🎯 Exam Focus
Write what the scale means before calculating. Show unit conversions clearly, keep full calculator values until the final step, and include the correct unit in the answer.
📌 One-Minute Summary
- A scale connects a drawing or model to the corresponding real object.
- Use multiplication when moving from a drawing measurement to a real measurement.
- Use division when working backwards from real size to drawing size.
- Ratio scales require the same unit on both sides.
- Area changes by the square of the linear scale factor.