Scaling Laws Math Example 1

Follow the full solution, then compare it with the other examples linked below.

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 ss, area s2s^2.
  2. 2
    Scaled square: side 3s3s, area (3s)2=9s2(3s)^2 = 9s^2.
  3. 3
    Area scales by 32=93^2 = 9.
  4. 4
    General law: scaling all lengths by factor kk scales area by k2k^2.

Answer

Area scales by k2=9 when lengths scale by k=3\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 k2k^2. This law is universal for any 2D shape, not just squares.

About Scaling Laws

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

Learn more about Scaling Laws →

More Scaling Laws Examples

Example 2 medium

A sphere of radius [formula] has volume [formula]. If the radius doubles, by what factor does the vo

Example 3 easy

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

Example 4 medium

A population model gives [formula]. If [formula] is doubled (growth rate doubles), by what factor do

← All Scaling Laws Examples Scaling Laws Concept