Equable Shapes

Date: 04/26/2001 at 14:16:33
From: Laura
Subject: Equable Shapes

I know equable shapes are shapes with the same area and perimeter, but 
can you give me any more info on the subject? Is there a rule for how 
to get equable shapes?

Date: 04/26/2001 at 16:47:30
From: Doctor Peterson
Subject: Re: Equable Shapes

Hi, Laura.

"Equable shape" is not a standard term, but seems to have been 
invented for the sake of this problem, which we get often. It's really 
very easy to work out; just write the formulas for area and perimeter 
of the given shape, which you can find in our Ask Dr. Math FAQ:

  http://mathforum.org/dr.math/faq/formulas/   

and set them equal, forming an equation. Then solve it. You should 
start with a simple case, such as a square:

      P = A
     4s = s^2

Solve that for s and you'll have an "equable" square. Then try doing 
the same for other shapes. You'll have to define each shape in terms 
of a single variable, so for instance you can't do this with a general 
rectangle (defined by two dimensions); but you can assume that the 
length-to-width ratio (or, if you prefer, the length itself) is known 
ahead of time, and treat only the width as a variable. Then the 
dimensions of the rectangle will be, say, s by ks, with k a constant, 
and you can write the area and perimeter formulas in terms of s and 
solve for s.

You should be aware that the concept of "equable shapes" is really 
nonsense. Area and perimeter can't really be equal, because they are 
measured in different units. Technically, we say they are "numerically 
equal," not actually "equal." Also, if you used different units (say, 
centimeters or furlongs instead of meters) to measure the figure, the 
area and perimeter would no longer be "equal." In fact, you can take 
ANY figure, and by choosing the right unit to measure it with, make 
the shape "equable" - so a better description of what you are doing 
would be to find the "equable size" or "equable unit" for a given 
shape.

I hope this is enough explanation to help you get started doing the 
work on your own. If you need more help, write back and show me what 
you were able to do.

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