The Square/Cube Rule: Volumes in Similar Solids
If two similar solids have a scale factor of , then surface areas scale by and volumes scale by .
Introduction
Past Knowledge
Similar figures (6.1). Scale factor. SA & Volume formulas (12.2–12.3).
Today's Goal
Apply the square-cube law: SA ∝ k², V ∝ k³.
Future Success
Physics (stress/strength scaling), biology (allometric growth), engineering.
Key Concepts
The Square-Cube Rule
| Measurement | Dimension | Scale Factor |
|---|---|---|
| Lengths | 1D | |
| Surface Area | 2D | |
| Volume | 3D |
Proof of the Rule
Consider two similar cubes with side lengths and :
SA₁ = 6s². SA₂ = 6(ks)² = 6k²s² = k² · SA₁ ✓
V₁ = s³. V₂ = (ks)³ = k³s³ = k³ · V₁ ✓
Since any solid can be approximated by tiny cubes, this holds for ALL similar solids.
∎ Area scales as the square of k, volume as the cube of k.
Worked Examples
Doubling Dimensions
You double every dimension of a box. How does the volume change?
k = 2. V scales by k³ = 2³ = 8 times.
Volume increases 8×
Finding New SA
Similar cylinders: radii 3 and 9. Small SA = 60π. Find large SA.
k = 9/3 = 3. SA scales by k² = 9. Large SA = 60π × 9 = 540π
SA = 540π
Volume Ratio → Scale Factor
Two similar solids have volumes 64 and 729. Find the ratio of their surface areas.
Volume ratio = 64:729. k³ = 64/729 → k = 4/9.
SA ratio = k² = 16/81.
SA ratio = 16:81
Common Pitfalls
Using k Instead of k² or k³
Lengths scale by k. Areas by k². Volumes by k³. The #1 mistake is applying the wrong power.
Real-Life Applications
Why Ants Can Carry 50× Their Weight
Muscle strength scales with cross-section (k²) but weight scales with volume (k³). Smaller creatures have more strength relative to weight. This is the square-cube law in biology.
Model Ships
A 1:100 scale model of a ship has 1/10,000 the surface area and 1/1,000,000 the volume of the real ship.
Practice Quiz
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