Making copies bigger or smaller
A scale factor tells how much a figure is enlarged or reduced. Every length is multiplied by it.
| Scale factor | Result |
|---|---|
| greater than 1 | enlargement (bigger) |
| exactly 1 | same size |
| between 0 and 1 | reduction (smaller) |
A rectangle 4 cm by 6 cm with scale factor 3 becomes 12 cm by 18 cm.
Finding a scale factor
Divide a length in the new figure by the corresponding side in the original:
scale factor = new length ÷ original length
Example: A 5 cm side becomes 2 cm. Scale factor = 2 ÷ 5 = 0.4 (a reduction).
Similar figures
Figures made with a scale factor are similar figures: they have the same shape, equal angles and proportional sides. Angles do not change with scale.
Finding a missing side
Example: Triangle A has sides 3, 4, 5. Triangle B is similar with longest side 20. Find B’s other sides.
- Scale factor: 20 ÷ 5 = 4
- Sides: 3 × 4 = 12, 4 × 4 = 16
Scale factor and perimeter
Perimeter scales by the same factor. A shape with perimeter 18 cm, scaled by 2.5, has perimeter 45 cm.
Scale factor and area
Area grows faster! If lengths multiply by k, area multiplies by k², the area factor.
| Scale factor | Area factor |
|---|---|
| 2 | 4 |
| 3 | 9 |
| 0.5 | 0.25 |
A 2 cm × 3 cm rectangle (6 cm²) enlarged by 3 becomes 6 cm × 9 cm (54 cm²): 9 times the area.
Volume of solid figures scales by k³.
Real-world scale factors
- Photocopiers: 150% means scale factor 1.5.
- Model cars: 1:24 means scale factor 1/24.
- Microscopes: 400× magnification.
Worked examples
Example 1: A 6 cm by 10 cm photo is enlarged by a scale factor of 1.5. New size?
9 cm by 15 cm.
Example 2: A side goes from 12 m to 3 m. Scale factor?
3 ÷ 12 = 0.25.
Example 3: A square of area 16 cm² is enlarged by scale factor 3. New area?
16 × 9 = 144 cm².
Common mistakes
- Dividing the wrong way (original ÷ new).
- Multiplying area by the scale factor instead of its square.
- Changing angles when scaling. Angles stay the same.
Try it yourself
- A 5 cm line is scaled by 0.6. New length?
- A side changes from 8 to 20. Scale factor?
- A triangle with perimeter 24 cm is scaled by 1/3. New perimeter?
- A rectangle with area 10 m² is scaled by 2. New area?
Answers: 1) 3 cm 2) 2.5 3) 8 cm 4) 40 m²
Quick recap
- Scale factor = new length ÷ original length.
- Greater than 1: enlargement; less than 1: reduction.
- Lengths scale by k, areas by k².