Ratio of Volume to Surface Area in Humans

Date: 12/09/1999 at 20:01:05
From: Frances Hall
Subject: Volume and surface area

Hi:

My students are studying giants and we are trying to determine the 
proper or healthy ratio of volume to surface area for humans. Can 
you help?

Date: 12/10/1999 at 09:19:50
From: Doctor Peterson
Subject: Re: Volume and surface area

Hi, Frances.

I don't know that there is a specific ideal ratio; and you'd probably 
want to ask doctors or medical researchers rather than mathematicians 
if there is. What I can tell you mathematically is that this ratio 
varies depending on both shape and size, being proportional to height 
if the shape is the same. A larger person of the same shape would 
have a greater ratio, which could cause overheating problems, so 
that it might be preferable to have a thinner shape to decrease the 
volume and improve cooling; but on the other hand the larger volume-
to-cross-sectional area ratio would cause excess stress on bones, 
making it preferable to have a thicker shape to increase the bone 
strength. In other words, I don't know that volume to area alone would 
give a good indication of health. But let me know if you find 
otherwise; it may be that within the normal size range this ratio does 
matter.

If you can find the book _Men, Ants, and Elephants: Size in the Animal 
World_ by Peter Weyl (1959), I would recommend it. It deals with 
exactly these issues in an entertaining way, explaining why elephants 
have thick legs, need big ears to help cool their bodies, etc., while 
ants have proportionally thinner legs, and don't even need lungs. It 
is a fascinating subject to study, and involves a nice combination of 
math and science.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/