Circumference vs. Perimeter

Date: 05/09/2003 at 12:01:23
From: Todd
Subject: Circumference vs. Perimeter

Technically speaking, can the term "perimeter" apply to a circle in a 
mathematical context?

While I would never actually refer to the circumference of a circle 
as its "perimeter," a discussion recently arose about whether the 
term "perimeter" can even APPLY to a circle. I consulted a standard 
dictionary, which said it was the length of a closed curve enclosing 
an area, so it would seem to apply under that definition. A 
mathematics dictionary said that perimeter was the sum of the lengths 
of the edges of a closed figure, but since a circle has no edges, 
this doesn't seem to apply to circles. However, the same mathematical 
dictionary defined circumference as "the perimeter of a circle."

Date: 05/09/2003 at 12:34:15
From: Doctor Peterson
Subject: Re: Circumference vs. Perimeter

Hi, Todd.

"Circumference" is just a special term for the perimeter when applied 
to circles. 

We have to have a general term that applies to all shapes, or it would 
get very confusing; there is no reason not to allow the word 
"perimeter" to be applied to circles, as part of a discussion that 
includes both circles and other shapes. For example, in the 
"isoperimetric problem" we are looking for the figure with the 
greatest area among all shapes with a given perimeter, and the answer 
turns out to be the circle. We'd be in bad shape if we weren't allowed 
to call that a perimeter, so that the circle was disqualified!

A dictionary that applies "perimeter" only to "edges" will we hope  
define "edge" in a way that is not limited to straight lines. I'm 
sure that's what was intended, since they just said "closed figure," 
not "polygon." In that sense, a circle has one edge.

See Eric Weisstein's World of Mathematics:

   Perimeter
   http://mathworld.wolfram.com/Perimeter.html 

which defines perimeter as the total arc length of a boundary, and 
specifically states that the perimeter of a circle is called the 
circumference. Your math dictionary is probably aimed at a less 
sophisticated audience, at least in that particular definition, which 
does not impress me much.

If you have any further questions, feel free to write back.

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

Date: 05/09/2003 at 12:43:53
From: Todd
Subject: Thank you (Circumference vs. Perimeter)

Thanks for the prompt response. I appreciate the reasoning behind your 
answer - makes sense to me! Kudos.