Relation between ratios of lengths, areas, and volumes

Statement

In terms of ratios

Suppose ${\displaystyle A}$ and ${\displaystyle B}$ are two objects of the same shape but possibly different size. Suppose the ratio of lengths in ${\displaystyle A}$ to corresponding lengths in ${\displaystyle B}$ is ${\displaystyle \lambda }$. Then:

• The ratio of areas in ${\displaystyle A}$ to corresponding areas in ${\displaystyle B}$ is ${\displaystyle \lambda ^{2}}$
• The ratio of volumes in ${\displaystyle A}$ to corresponding volumes in ${\displaystyle B}$ is ${\displaystyle \lambda ^{3}}$

Note that even if ${\displaystyle A}$ and ${\displaystyle B}$ do not have exactly the same shape, the above still holds in an approximate sense as long as the shapes are reasonably similar.

In terms of logs of ratios, or orders of magnitude

Suppose ${\displaystyle A}$ and ${\displaystyle B}$ are two objects of the same shape but possibly different size. Suppose that lengths in ${\displaystyle A}$ are ${\displaystyle m}$ orders of magnitude greater than corresponding lengths in ${\displaystyle B}$. Then:

• Areas in ${\displaystyle A}$ are ${\displaystyle 2m}$ orders of magnitude greater than corresponding areas in ${\displaystyle B}$.
• Volumes in ${\displaystyle A}$ are ${\displaystyle 3m}$ orders of magnitude greater than corresponding volumes in ${\displaystyle B}$.

Note that even if ${\displaystyle A}$ and ${\displaystyle B}$ do not have exactly the same shape, the above still holds in an approximate sense as long as the shapes are reasonably similar.

Formally, ${\displaystyle m}$ is the logarithm of ${\displaystyle \lambda }$ (to base 10 if we are describing orders of magnitude in terms of powers of 10).