Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
Enlarging a shape means multiplying all its lengths by the same number k, called the scale factor. If k is bigger than 1, the shape grows; if k is between 0 and 1, it shrinks: that is a reduction.
Trick question: if all the lengths are multiplied by k, what happens to the area? Take a square with side 2 cm, area 4 cm². Enlarge it with k = 3: the side becomes 6 cm, and the area... 36 cm²! The area was not multiplied by 3, but by 9, that is by 3².
The reason is illuminating: an area is two lengths multiplied together. Each one is multiplied by k, so the area is multiplied by k × k = k². In the same way, a volume combines three lengths: it is multiplied by k × k × k = k³.
Remember the trio: lengths × k, areas × k², volumes × k³. With k = 2: lengths double, areas are multiplied by 4, volumes by 8. That is why a balloon twice as wide holds eight times more air!
These rules are used everywhere: maps and plans, scale models, enlarged photocopies, nesting dolls. A plan at scale 1/100 divides all real lengths by 100: and on that plan, real areas are divided by 100² = 10000!
We enlarge a square with side 2 cm by the factor k = 3. What do its side and its area become?
côté : 2 × 3 = 6
Lengths are multiplied by k: the side goes from 2 cm to 6 cm.
aire avant : 2 × 2 = 4
The starting area is 4 cm².
aire après : 6 × 6 = 36
The new area is 36 cm²: it was multiplied by 9, that is by k² = 3².
Result: Side × 3, but area × 9: areas follow k².
A scale model reduces a solid with k = 1/2. The real volume is 40 cm³. What is the model's volume?
volumes : × k³
Volumes are multiplied by k cubed, because three dimensions shrink at the same time.
k³ = 1/2 × 1/2 × 1/2 = 1/8
With k = 1/2, the factor on volumes is 1/8.
40 ÷ 8 = 5
The model's volume is 40 divided by 8, that is 5 cm³.
Result: The model's volume is 5 cm³.