Caption: A scaling law (AKA power law) holds when one variable grows as a power of an another variable.

In the image, we see that the area of a square scales as the 2nd power (i.e., the square) of the length of an edge and the volume of a cube scales as the 3rd power (i.e., the cube) of the length of an edge.

In fact, any shape in a plane scales as the 2nd power of any characteristic length because any shape can be imagined as being constructed of infinitesimal squares. Similarly, any shape in 3-dimensional space scales as the 3rd power of any characteristic length because any shape can be imagined as being constructed of infinitesimal cubes.

In physics, extensive quantities are those that scale with volume of a system as you magically scale the system up or down: e.g., volume itself, entropy, heat energy, mass, etc.

On the other hand, intensive quantities are unchanged by magical scaling usually because they are ratios of extensive quantities: e.g., density, pressure, specfic heat capacity, etc.

Temperature
is another
intensive quantity,
but it is usually **NOT** thought
of as a
ratio of
extensive quantities
though it can be in some senses
(see Wikipedia: Temperature:
Theoretical foundation).

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User:LaurensvanLieshout,
2009 /
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CC BY-SA 3.0.

Image link: Wikimedia Commons:
File:Scaling law.png.

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