Comparing dimensions: linear, area, volume
When two shapes are similar (same shape, different size), all corresponding lengths share a common scale factor. But areas and volumes scale by powers of that factor.
The three scale factors
If linear scale factor (LSF) is k then:
- Area scale factor (ASF) = k².
- Volume scale factor (VSF) = k³.
So if a model is half the size of the real thing (k = 1/2):
- Areas are 1/4 of the real.
- Volumes are 1/8 of the real.
✦Worked example— Worked examples
Example 1. Two similar triangles have lengths in ratio 2 : 3. What is the ratio of their areas?
- ASF = 2² : 3² = 4 : 9.
Example 2. Two similar cones have surface areas 16 cm² and 25 cm². Find the ratio of their heights.
- ASF = 16 : 25 → LSF = 4 : 5.
- Heights are in ratio 4 : 5.
Example 3. Two similar containers have volumes 27 ml and 125 ml. The smaller has surface area 36 cm². Find the surface area of the larger.
- VSF 27 : 125 → LSF 3 : 5 → ASF 9 : 25.
- 36 × 25/9 = 100 cm².
Setting up the right ratio
When comparing, write the smaller : larger consistently to avoid sign mix-ups.
Why areas scale by k²
A square of side L has area L². Doubling the side gives a 2L × 2L = 4L² square — four times the area. The "linear factor" of 2 became the area factor of 4 because of the two-dimensional measure.
Why volumes scale by k³
A cube of side L has volume L³. Doubling sides gives V = (2L)³ = 8L³ — eight times.
⚠Common mistakes
- Linear ratio used directly for area — 2 : 3 is NOT the area ratio.
- Ratio direction flip mid-question — pick smaller : larger and stick with it.
- Mixing area scale with volume scale — square root vs cube root direction.
- Forgetting to take the square / cube root when working from ASF or VSF back to LSF.
➜Try this— Quick check
Two similar bottles have heights 8 cm and 12 cm. The smaller has volume 200 ml. Find the volume of the larger.
- LSF: 8:12 = 2:3. VSF: 8:27.
- 200 × 27/8 = 675 ml.
AI-generated · claude-opus-4-7 · v3-deep-ratio