Dividing Land by Area vs. Perimeter

Date: 11/05/1999 at 10:23:04
From: mark werner
Subject: Area versus Perimeter

I'm trying to determine what the advantages are of dividing land by 
area versus perimeter. I think area is better because there are fewer 
options.

Date: 11/05/1999 at 12:22:53
From: Doctor Peterson
Subject: Re: Area versus Perimeter

Hi, Mark.

I'm not quite sure what you would mean by "dividing land via 
perimeter." If I were to inherit some land that I had to divide with 
my brothers, of course I'd want each of us to get the same area, 
because that's usually what determines its value. If we split it so we 
each had the same perimeter, we'd each have to build the same amount 
of fence, but that wouldn't matter much if my piece looked like this

     +-------------------------------------------+
     |                                           | 1
     +-------------------------------------------+
                           43

and my brother's looked like this

     +----------------------+
     |                      |
     |                      |
     |                      |
     |                      |
     |                      | 22
     |                      |
     |                      |
     |                      |
     |                      |
     |                      |
     +----------------------+
                22

There would be a big difference in how much land we had; my fence 
wouldn't hold much.

Maybe you can give me the exact wording of the original problem, and a 
description of whatever picture there might be with it.

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

Date: 11/08/1999 at 08:59:01
From: Mark Werner
Subject: Re: Area versus Perimeter

In the question, we have a piece of land that has a perimeter of 180'. 
We have to recommend to the city council one of two options. Should we 
assign land to the homeowners by area or by perimeter? Then we have 
to determine what the pros and cons are of each recommendation. I'm 
favoring the area option. My conclusion is to divide the land into a 
45 by 45 square. This maximizes my area available with a 180' 
perimeter (45*45 = 2025). I'm not sure what other pros there are to 
this option, or what would be the pros for allocating land by 
perimeter.

Date: 11/09/1999 at 08:41:33
From: Doctor Peterson
Subject: Re: Area versus Perimeter

Hi, Mark.

I'm still missing one thing: what would it MEAN to divide, or 
allocate, land by perimeter? I can't think HOW you'd do it, much less 
WHY. I assume the idea would be to give each person a part of this 
parcel that has the same perimeter. (Though the parcel probably isn't 
big enough for more than one family anyway - and it could be 89 feet 
by one foot, for all I know!)

If you knew the dimensions of the plot, say 40 by 50 feet, and you 
were allowed to put a fence in it dividing your "half" from someone 
else's, wherever you wanted as long as you both had the same 
perimeter, you could do this:

     +-------------------+-------------------+
     |                   |                   |
     |                   +---------------+   |
     |                                   |   |
     |                                   |   |
     |                                   |   |
     |                                   |   |
     |          Mine                     |His|
     |                                   |   |
     |                                   |   |
     |                                   |   |
     |                                   |   |
     |                   +---------------+   |
     |                   |                   |
     +-------------------+-------------------+

You both have the same perimeter, but you have most of the land. Such 
a "division" would be meaningless.

It is easy to do this for two people, because they share the whole 
added border. If you tried to do this for three people, it would be a 
very difficult problem to solve just to make the perimeters come out 
equal, because the perimeters of the parts don't add up to the 
perimeter of the whole, as areas do. You wouldn't know until everyone 
had chosen his or her piece whether you had chosen a perimeter that 
everyone else would have. With areas, you would know you had to take 
1/3 of the total area, but there's no limit on the perimeters of the 
pieces.

But what you've said doesn't seem to involve dividing a given piece of 
land; what you've done is only to find the shape of a piece with that 
perimeter that has the maximum area. Where does "dividing" or 
"allocating" come into this? And it sounds as if you're already 
allocating by perimeter - you're taking 180 feet of perimeter for 
yourself, and then choosing the best shape on the basis of area. Is 
the idea really that there's a lot of free land and they want to 
decide whether to give everyone a piece of it with 180 feet perimeter, 
or one with, say, 2025 square feet?

I'm still baffled!

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