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### Equal Area and Perimeter: Rectangles

```
Date: 09/09/2001 at 13:56:35
From: Jessica
Subject: Area and perimeter

Hi!

There are only two rectangles whose area is exactly the same as their
perimeter if the dimensions of each are whole numbers. What are the
dimensions?

Thanks,
Jessica
```

```
Date: 09/10/2001 at 12:46:10
From: Doctor Ian
Subject: Re: Area and perimeter

Hi Jessica,

Let's look at a rectangle for a moment:

a
+------------+
|            |
|            | b
+------------+

The area will be a*b; the perimeter will be 2a + 2b.  So we want to
find pairs of numbers (a,b) such that

a*b = 2a + 2b

More specifically, we want to find pairs of whole numbers for which
this equation is true.  One simple way to attack a problem like this
is to start making a table:

a    b    a*b   2a+2b
---  ---   ---   -----
0    0     0      0
0    1     0      2
1    0     0      2

So we've found one solution already.  A 0 by 0 rectangle has an area
equal to its perimeter.

(This seems kind of strange, I know, but it fits the definition. And
note that the problem explicitly says that the numbers must be 'whole
numbers', not 'counting numbers'.  The only difference between
counting numbers and whole numbers is that the whole numbers include
zero. So whenever you see a problem that talks about 'whole numbers',
that's a tip that zero might turn out to be important.)

Now, the tough part about constructing a table like this is to make
sure that you cover all the possibilities. How can we guarantee that
we get all the possible (a,b) pairs?  What if we miss one, and that
turns out to be the answer? We could keep looking forever.

Well, here is a trick for ordering these pairs:

|
10  11   12   13
|
5    6    7   14
|
2    3    8   15
|
1----4----9---16---

The coordinates of the points, in order, are:

1:  (0,0)

2;  (0,1)
3:  (1,1)
4:  (1,0)

5:  (0,2)
6:  (1,2)
7:  (2,2)
8:  (2,1)
9:  (2,0)

10:  (0,3)
11:  (1,3)

and so on. Do you see why this will get _every_ pair of whole numbers?

Of course, it will get some of them twice (for example, (2,1) and
(1,2) actually describe the same rectangle), which creates some extra
work. (I included both (0,1) and (1,0) in the table above before I
realized this. I didn't need to work out both of them.)

If we want to be lazy (which in math is often a good thing), we can
just look at the points above the diagonal:

|
7    8    9   10
|
4    5    6
|
2    3
|
1------------------

(Do you see why this works?)

If we want to be even lazier, we can note that since

a*b = 2a + 2b

= 2(a + b)

the product a*b must be an even number. This is a break, because it
means that we can ignore any (a,b) pair in which both a and b are odd.
So we can forget about possibilities like (1,3), or (3,5), since there
is no way to multiply two odd numbers to get an even number.

Anyway, now you can make a table, which will let you find the other

Now, when making a table like this, there are two possibilities:

1. You'll run across the answer very quickly. If that happens,
then using a table will turn out to have been a good idea.

2.  You won't run across the answer very quickly.  If that happens,
there are two more possibilities.

a.  You'll notice some pattern in the table that will
having to construct all the table entries in between.
If that happens, then using a table will turn out to
have been a good idea.

b. You won't notice any pattern like that. In that
case, you'll want to start looking around for another
way to solve the problem.

Fortunately, in this case, you should come across the answer pretty
quickly, if you use both of the shortcuts that I mentioned.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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