Find the Perimeter of the Fence

Date: 02/19/2003 at 12:13:04
From: Dzevdan Kapetanovic
Subject: Geometry

A convex polygon with perimeter P has a fence around it. The distance 
from the polygon to the fence is 1m. How long is the fence?

It's hard to get a geometrical picture to work with. The sides of the 
fence must be parallel to the sides of the polygon, since distance is 
defined to be the orthogonal line between two sides. So the problem is 
the edges. We don't know what angle an edge has, because it seems the 
fence must be curved along the edges (an arc with radius 1) so that 
the distance can be 1 m. Or is it just enough that the distance from 
the edge of the fence to the edge of the polygon is 1 m?

Best regards.

Date: 02/19/2003 at 16:59:06
From: Doctor Peterson
Subject: Re: Geometry

Hi, Dzevdan.

Let's draw an arbitrary polygon to illustrate what is happening:

               ..|...........|..
              \  |           |  .
             .   A-----------B----
            .   /            |   .
           .   /             |   .
          .   /              |   .
         .   /               |   .
        .   /                |   .
       .   /                 |   .
      .   D                  |   .
        /  \                 |   .
      /.     \               |   .
         .     \             |   .
           .     \           |   .
             .     \         |   .
               .     \       |   .
                 .     \     |   .
                   .     \   |   .
                     .     \ |   .
                       .     C-----
                         . /    .
                         / .....

Here I have drawn a quadrilateral ABCD, with a "fence" around it one 
unit away. I have interpreted this as you did, so that the fence is 
parallel to the edges where possible, and makes an arc with radius 1 
unit at the corners. I have drawn the radii at the start and end of 
each arc, which of course are perpendicular to the neighboring edge 
of the polygon.

Note that I have thus decomposed the region between the polygon and 
the fence into rectangles and circular sectors. Now consider what 
happens if you put all the sectors together into one circle, 
essentially collapsing the polygon into a point by reducing each edge 
to nothing:

       ..|..
      \  |  .
     .   +----
      ./    .
     / .....

Can you see how you can now find the perimeter of the fence, without 
having to know anything about the polygon except that it is convex, 
and what its perimeter is?

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

Date: 02/20/2003 at 18:59:51
From: Dzevdan Kapetanovic
Subject: Geometry

Hello Dr. Peterson!

I am very thankful for an answer!

It's hard for me to see the last picture exactly, and my English 
isn't quite the best, but as I understand it, you formed a circle with 
those arcs that are around every edge in the polygon. That's 
possible because you get a fence that consists of straight sides and 
arcs. Then if you let the sides go to 0, you will get a circle that
will certainly have a radius of 1 unit. Then the length of those arcs
is 2*pi, and the sum of the lengths of those sides of the fence, 
parallel to the polygon sides, are certainly P. So then the total 
length would be P + 2*pi right? Is that correct?

That's what I got before, too (the first time I wrote to you), but it 
seems like I had the wrong argument. I assumed that if you formed a 
circle of the fence and a circle of the polygon then the distance 
would still be 1 unit between them and certainly they would still 
have same lengths.

Grateful for an answer

Best regards

Kapetanovic

Date: 02/20/2003 at 22:02:36
From: Doctor Peterson
Subject: Re: Geometry

Hi, Dzevdan.

Yes, you understood what I meant, and your answer is correct.

It would be difficult to prove your assumption that the length of the 
fence would be the same regardless of the shape of the polygon, but 
having shown that it is true for any convex polygon, it does appear 
that it is true. You could form regular polygons with more and more 
sides, and in the limit the length of the fence around a circle would 
still be P + 2 pi. This reminds me of the well-known problem that 
asks how much wire would have to be added to a cable around the world 
if you raised it one meter off the ground; or, conversely how high it 
would rise if you added 6 meters to its length.


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