If the lengths of two similar solids are in the ratio 1 : k, then their surface areas are in the ratio 1 : k² and their volumes are in the ratio 1 : k³. Volume is three-dimensional, so the scale factor is used three times.
You will need both the linear scale factor and the area relationship before this lesson.
Why is volume multiplied by k³?
Take a cuboid 2 cm by 3 cm by 4 cm. Its volume is 24 cm³. Double every length and it becomes 4 cm by 6 cm by 8 cm, with volume 192 cm³.
Now 192 ÷ 24 = 8, and 8 = 2³. Each of length, width and height is scaled by 2, so the volume is scaled by 2 × 2 × 2.
The pattern is one power of k for each dimension: length k, area k², volume k³.
The method
- Find the linear scale factor k from a pair of corresponding lengths.
- Decide what you are scaling: a length, an area (surface area), or a volume (or capacity, mass of the same material).
- Use the right power: k, k² or k³.
- If you start from volumes, take the ratio, simplify, then take the cube root to get k.
Worked example
Two similar bottles have heights 12 cm and 18 cm. The smaller bottle holds 240 ml. How much does the larger hold?
Step 1, scale factor. 18 ÷ 12 = 1.5.
Step 2, volume factor. 1.5³ = 1.5 × 1.5 × 1.5 = 3.375.
Step 3, capacity. 240 × 3.375 = 810 ml.
Step 4, check. 240 × 3 = 720 and 240 × 0.375 = 90, and 720 + 90 = 810.
Starting from volumes. Two similar solids have volumes 54 cm³ and 250 cm³. The larger is 20 cm high. The volume ratio is 250 ÷ 54 = 125/27, so k = 5/3 from small to large. The smaller solid is 20 × 3/5 = 12 cm high. The surface area ratio would be (5/3)² = 25/9.
The mistake to watch for
A common slip is to use the wrong power, usually the square, because area was the previous lesson.
Mistaken working: 240 × 1.5² = 240 × 2.25 = 540 ml.
The student used the area factor for a capacity. A volume has three dimensions, so the factor must be cubed.
The correction is to ask what kind of quantity the question asks for. Capacity, volume and mass of the same material use k³, surface area uses k², and length uses k.
Write the power beside the quantity before you calculate.
Check yourself
Try these, then open each answer.
1. A solid has volume 35 cm³. It is enlarged with scale factor 2. Find the new volume.
Show answer
Volume factor = 2³ = 8. New volume = 35 × 8 = 280 cm³.
2. Two similar solids have volumes in the ratio 8 : 27. The surface area of the smaller solid is 50 cm². Find the surface area of the larger solid.
Show answer
The cube root of 8/27 is 2/3, so the length ratio is 2 : 3. The area ratio is 4 : 9. Going from the smaller to the larger, the factor is 9/4 = 2.25. Surface area = 50 × 2.25 = 112.5 cm².
3. A model of a tank is built at scale 1 : 10. The model has volume 64 cm³. Find the real volume in litres. (1 litre = 1000 cm³.)
Show answer
Volume factor = 10³ = 1000. Real volume = 64 × 1000 = 64 000 cm³. Dividing by 1000 gives 64 litres.
Where this leads next
You now have the full set of length, area and volume factors. Test them mixed together in the similarity, congruence and scale practice set, or first look at how similarity differs from congruence. The map scale and scale-factor practice tool gives extra repetitions.
Mixed questions are where the wrong power creeps in. A teacher in online one-to-one Mathematics tuition can work through several with you and build a labelling habit that stops it.