Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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/   
    
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/