Lesson Guide
Explanation
When you enlarge a shape by a scale factor k, every length is multiplied by k, but the area is not. Look at the rectangle. Its length goes from two to six, and its width goes from one to three, so the scale factor is three. The new rectangle holds nine copies of the old one: three across and three down. So the area is multiplied by nine, which is three squared. In general, lengths are multiplied by k and areas are multiplied by k squared.
Your Turn
A shape of area 5 cm2 is enlarged by k = 4. Check: area × 16, so the new area is 80 cm2.
Watch Out
Area does not grow by k. Tripling the lengths makes the area 9 times as big, not 3 times.
Explanation
Volume has three dimensions, so it is multiplied by k three times. This box measures two by one by one, so its volume is two. Enlarge it by a scale factor of two and it measures four by two by two, with volume sixteen. Sixteen divided by two is eight, which is two cubed. So lengths are multiplied by k, areas by k squared, and volumes by k cubed. Surface area, like the paint on the box, is an area, so it goes up by k squared, which here is four.
Your Turn
A solid of volume 3 cm3 is enlarged by k = 3. Check: volume × 27, so 81 cm3.
Watch Out
Volume is k3, not k2. Doubling every length makes a box hold 8 times as much, not 4 times.
Explanation
Sometimes you know how the areas compare and you need the lengths. Since areas are lengths squared, go backwards by taking the square root. If two similar shapes have areas in the ratio nine to twenty five, square-root each part: the lengths are in the ratio three to five. Now you can go forwards to anything else. Cube three and five, and the volumes of two similar solids with that area ratio are in the ratio twenty seven to one hundred and twenty five.
Your Turn
Areas 4 : 49. Check: lengths 2 : 7, so volumes 8 : 343.
Watch Out
Do not use the area ratio for lengths. With areas 9 : 25, a 6 cm side matches 10 cm (×\(\dfrac{5}{3}\)), not 6 × \(\dfrac{25}{9}\).
Explanation
Volumes are lengths cubed, so to go backwards from a volume ratio you take the cube root. Two similar jars have volumes in the ratio twenty seven to sixty four. The cube root of twenty seven is three and the cube root of sixty four is four, so their heights, and all other lengths, are in the ratio three to four. Square that to get the ratio of their surface areas: nine to sixteen. That tells you how much more paint or label the bigger jar needs.
Your Turn
Volumes 1 : 125. Check: lengths 1 : 5, so surface areas 1 : 25.
Watch Out
Cube-root a volume ratio; do not square-root it. √27 : √64 is not a length ratio, and it is not even a pair of whole numbers.
Explanation
Models are scale drawings in three dimensions. A model water tank is built to a scale of one to ten, so every real length is ten times the model's. The model needs zero point two litres of paint. Paint covers the surface, which is an area, so multiply by ten squared, which is one hundred: the real tank needs twenty litres. The model holds three litres. Capacity is a volume, so multiply by ten cubed, which is one thousand: the real tank holds three thousand litres. Quiz and exercises now.
Your Turn
A 1 : 4 model needs 50 mL of paint and holds 0.5 L. Check: paint 50 × 16 = 800 mL; capacity 0.5 × 64 = 32 L.
Watch Out
Paint is NOT volume. Multiplying the paint by 103 gives 200 L, which is far too much.
📌 Key Points — this lesson in a nutshell
- If every length is multiplied by k, every area is multiplied by k2 and every volume by k3.
- Length ratio a : b gives area ratio a2 : b2 and volume ratio a3 : b3.
- Working backwards: square-root an area ratio, or cube-root a volume ratio, to get the length ratio. Area 9 : 25 → lengths 3 : 5.
- Paint and surface area go with k2. Capacity, mass of solid material and volume go with k3.
- Convert area and volume units with the same rule: 1 m2 = 1002 cm2 = 10 000 cm2, and 1 m3 = 1 000 000 cm3.
Step 1: Introduction
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