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If You Know Perimeter, Can You Find Area?

Date: 6/30/96 at 11:42:8
From: Clay A Thompson
Subject: Given Perimeter, Find Area?

Can one determine the acreage of an irregularly shaped field if only 
the distance around the edge of the field (in feet) is known?

Thanks for your help.

Date: 6/30/96 at 14:33:8
From: Doctor Ceeks
Subject: Re: Given Perimeter, Find Area?

Unfortunately, the perimeter of a region does not determine the 
region's area.

In fact, for a given perimeter P, the area of the region can be 
anything between 0 and P^2/(4 pi).  The maximum can be achieved only 
by using a circular region.

-Doctor Ceeks,  The Math Forum
 Check out our web site!   

Date: 11/19/2010 at 11:32:18
From: Gerard
Subject: area of irregular field from known perimiter

On this web page, you state that finding the area of an irregular field is
impossible given just the perimeter because the area could be anywhere
from 0 to the area of a circle.

Your answer defies physics and common sense in the real world in which the
question was framed.

A field with a measured perimeter cannot have an area of zero; and while
there are growing areas that are perfect circles (thanks to rotary
irrigation), the original post was about an irregular field.

While one cannot ascertain the exact area of the stated field, one can
come to an answer that will be close to a correct answer with some margin
of error. This is often the best we can do when dealing with real-world

I think the proper answer would have been to explain how to convert the
perimeter's length into a square, triangle, or other simple three- or
four-sided shape and solve for that.

Date: 11/19/2010 at 23:12:40
From: Doctor Peterson
Subject: Re: area of irregular field from known perimiter

Hi, Gerard.

To some extent, you may be right; if the question does not need an exact
answer, one could estimate what the area MIGHT be by making a few
assumptions. Often we get questions from people who describe a lot as a
"rectangle," but then give dimensions that show it is not really a
rectangle. In those cases, we do as you suggest, supposing the shape is
close enough to a rectangle, and then working out an approximate area
without worrying about the details. (On the other hand, if they are asking
for tax purposes or something like that, an actual survey would be needed,
not just a broad estimate.)

The big problem comes when we're only told it's "irregular," with no
details at all. Some fields are long and narrow, others are close to
square, some have lots of ins and outs; the result will be a major error
if we just guess. SOME information about the shape is needed if you want
something better than "the MOST it can be is X, the area of a circle with
that circumference," as Dr. Ceeks said.

For example, suppose we had two fields, one square and the other 4 times
as long as it is wide (which is probably not at all unusual). If they both
have the same perimeter P, then the square has sides P/4 and area
(P^2)/16, while the rectangle has sides P/10 and 2P/5, which gives area
(P^2)/25. The ratio of the areas is 16:25, or 0.64:1. For shape that are
not so "irregular," that's not a small margin of error!

If we had, say, a square and an "L" shape with the same perimeter, the
areas could be wildly different:

   +------------+    +---+
   |            |    |   |
   |            |    |   |
   |            |    |   |
   |            |    |   +--------+
   |            |    |            |
   +------------+    +------------+

So I think Dr. Ceeks gave the only possible answer to the question as
stated, short of asking "Can you describe the field so we can get some
idea what shape it approximates?" He was asked for a general rule, and the
answer is that there is none.

- Doctor Peterson, The Math Forum
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
Middle School Geometry
Middle School Two-Dimensional Geometry

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