Scaling Laws Math Example 3

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

easy

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

1
Circumference $C = 2\pi r$ : if $r \to 2r$ , then $C \to 2\pi(2r) = 2C$ . Circumference doubles.
2
Area $A = \pi r^2$ : if $r \to 2r$ , then $A \to \pi(2r)^2 = 4\pi r^2 = 4A$ . Area quadruples.

Answer

\text{Circumference} \times 2,\quad \text{Area} \times 4

Circumference scales linearly (as

k^1

) while area scales quadratically (as

k^2

). This is because circumference is a 1D measurement and area is 2D.

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.

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