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### Equable Shapes

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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

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/
```
Associated Topics:
Elementary Definitions
Elementary Geometry
Elementary Triangles and Other Polygons
Elementary Two-Dimensional Geometry
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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