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A Rectangular Prism

Date: 11/26/95 at 14:31:1
From: Anonymous
Subject: rectangular prism whose volume is greater than or equal to 
its surface area

I have been doing a Geometry Scavenger hunt for school, and I have 
been pondering this for days, is it possible to have a rectangular 
prism that has a volume greater than it's surface area?
I have tried to everything I can think of and I can't figure it out!
If it is possible could you please give me the measurements!

Date: 11/27/95 at 9:5:9
From: Doctor Ian
Subject: Re: rectangular prism whose volume is greater than or equal to 
its surface area 


Strictly speaking, it doesn't make sense to say that a volume is greater 
than an area, any more than it makes sense to say that a weight is greater 
than a length.  Volume and area are measures of different kinds of things, 
so they're not directly comparable.

However, we can ask the question in this way:  Is it possible to have 
a rectangular prism such that if V is the volume measured in cubic 
[whatever]s, and A is the surface area measured in square [whatever]s, 
then V > A? 

Asked this way, the answer is 'yes'.  For example, suppose we have a cube 
that is 1 foot on each edge.  The surface area is 

   A = 6 * (1 ft * 1 ft)

     = 6 square feet

while the volume is 

   V = (1 ft * 1 ft * 1 ft)

     = 1 cubic foot

Clearly, 6 > 1.  On the other hand, if we measure lengths in inches 
rather than feet, the area is

   A = 6 * (12 in * 12 in)

     = 864 square inches

while the volume is 

   V = (12 in * 12 in * 12 in)

     = 1728 cubic inches

Clearly, 864 < 1728. So which is 'larger', the surface area or the volume? 
We haven't changed the cube at all, just the units we use to measure length,
which are arbitrary. 

If the relative size depends on which units we choose, can the question of
which is larger really make sense? 

-Doctor Ian,  The Geometry Forum

Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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