# Relation between ratios of lengths, areas, and volumes

## Statement

### In terms of ratios

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

• The ratio of areas in $A$ to corresponding areas in $B$ is $\lambda ^{2}$ • The ratio of volumes in $A$ to corresponding volumes in $B$ is $\lambda ^{3}$ Note that even if $A$ and $B$ 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 $A$ and $B$ are two objects of the same shape but possibly different size. Suppose that lengths in $A$ are $m$ orders of magnitude greater than corresponding lengths in $B$ . Then:

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

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

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