Perimeter of a Line

Date: 08/25/2002 at 19:06:07
From: Teresa Lara-Meloy
Subject: Perimeter of a line

Hi,

I've been working with a small group of college kids in Mexico on a 
problem dealing with fixed-perimeter and variable area of rectangles.  
We were using a rectangle with perimeter 36cm. I asked them to graph 
perimeter, areas, etc. Then one of the students said "for a height of 
zero, then, I will need 18 at the base" ... which made sense according 
to their graph. She was challenged by other students who argued that a 
line of 18 has only perimeter of 18 - that she would need a line of 
36 units to get a perimeter of 36 units. So another student posed the 
question: "Does a line have perimeter?"  

We've had lots of discussion on this topic, but I donīt have a good 
answer given all their comments. Can you help?

Thank you very much!
Teresa

Date: 08/25/2002 at 19:42:17
From: Doctor Peterson
Subject: Re: Perimeter of a line

Hi, Teresa.

I would say that a line (segment) does not have a perimeter, since 
that term is applied only to polygons, or, more generally, to closed 
curves. The segment is a sort of degenerate rectangle, the limiting 
case when one dimension of a rectangle takes on the illegal value of 
zero. It would be best to see this, therefore, in terms of limits: as 
the height approaches zero, the area approaches zero while the 
perimeter remains 36; so although the limit does not properly have a 
perimeter, we can define its perimeter as twice the width, so that it 
is still 36. From this perspective, we can think of the segment as a 
rectangle with height zero and top and bottom edges (which coincide) 
18 cm long. But apart from this context, it would not make sense to 
talk about the perimeter. This is something like the concept of 0/0 
as an indeterminate form; in itself, it is not defined, but if you 
consider it as the limit of a particular ratio, it can be given a 
definition for that specific case.

If you were asked, "what is the smallest area of any rectangle with 
perimeter 36 cm?" you would have to say there is no such minimum, 
since the figure with area zero is not an actual rectangle, and any 
positive height gives a small positive area, which you can make as 
small as you want.

Does that help?

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