Isoperimetric Quotient

Date: 03/11/2002 at 09:20:40
From: Andreas Mantius
Subject: Surface Area to volume ratio

I have a question about a ratio equation of surface area to volume. I 
can't remember if its SA/V or V/SA.

Andreas

Date: 03/11/2002 at 12:38:34
From: Doctor Peterson
Subject: Re: Surface Area to volume ratio

Hi, Andreas.

The ratio of surface area to volume is by definition A/V; but that is 
not a meaningful quantity, because it depends on the units you use. A 
measure of "efficiency" that you might have heard of for plane figures 
is the isoperimetric quotient:

   Isoperimetric Quotient - MathWorld, Eric Weisstein
   http://mathworld.wolfram.com/IsoperimetricQuotient.html   

   Area and Perimeter: Isoperimetric Quotient - Dr. Math archives
   http://mathforum.org/dr.math/problems/ace.12.05.01.html   


    Q = 4 pi A / P^2

which for a circle is

    Q = 4 pi (pi r^2) / (2 pi r)^2 = 1

The equivalent for a solid figure, which I haven't heard of, would be 
an "isoareal quotient"

    Q = 36 pi V^2 / S^3

which I chose so that for a sphere,

    Q = (36 pi) (4/3 pi r^3)^2 / (4 pi r^2)^3 = 1

At any rate, since area is measured in square units and volume in 
cubic units, the ratio of the square of the volume to the cube of the 
surface area is what you need to eliminate dependence on units and 
consider only the shape itself.

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