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/
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