Scaling Laws Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling Laws.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Relationships describing how a quantity changes when the size or scale of a system is multiplied by a factor, often expressed as power laws.

When you double the length of a cube, its volume grows by 2^3 = 8. Scaling laws reveal how fast quantities grow — they often explain why small and large things behave so differently.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Scaling reveals why things work at one size but not another.

Common stuck point: Different quantities scale differently (linear, quadratic, cubic).

Sense of Study hint: Replace each length L with 2L in the formula and simplify. The factor that appears (2, 4, 8, etc.) tells you whether the quantity scales linearly, quadratically, or cubically.

Worked Examples

Example 1

easy
If you scale a square's side length by a factor of 3, by what factor does the area change? State the general scaling law.

Solution

  1. 1
    Original square: side s, area s^2.
  2. 2
    Scaled square: side 3s, area (3s)^2 = 9s^2.
  3. 3
    Area scales by 3^2 = 9.
  4. 4
    General law: scaling all lengths by factor k scales area by k^2.

Answer

\text{Area scales by } k^2 = 9 \text{ when lengths scale by } k=3
Scaling laws describe how quantities change when a single scale factor is applied. Area is a 2-dimensional quantity, so it scales as k^2. This law is universal for any 2D shape, not just squares.

Example 2

medium
A sphere of radius r has volume V = \frac{4}{3}\pi r^3. If the radius doubles, by what factor does the volume increase? Generalise to a scaling factor k.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
If the radius of a circle doubles, by what factor does the circumference change? What about the area?

Example 2

medium
A population model gives P(t) = P_0 e^{rt}. If r is doubled (growth rate doubles), by what factor does P(T) change for a fixed time T?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

dimensional reasoningproportionality