Relation between ratios of lengths, areas, and volumes

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Statement

In terms of ratios

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

  • The ratio of areas in to corresponding areas in is
  • The ratio of volumes in to corresponding volumes in is

Note that even if and 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 and are two objects of the same shape but possibly different size. Suppose that lengths in are orders of magnitude greater than corresponding lengths in . Then:

  • Areas in are orders of magnitude greater than corresponding areas in .
  • Volumes in are orders of magnitude greater than corresponding volumes in .

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

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