Equable Shapes: Triangle

Date: 08/27/2001 at 06:53:55
From: Natalie Hodson
Subject: Equiable Shapes

Hi,

I have been asked to do a piece of coursework on equiable shapes. I 
listened carefully to the teacher and asked her to repeat it, but I 
still don't understand. I have the square number, but no other one. I 
am so stuck on the triangle it is unreal. I have no idea how to find 
the number to make it equiable and no idea what to do with it from 
ten on. Any help would be greatly appreciated.

Thanks,

Natalie

Date: 08/27/2001 at 08:51:49
From: Doctor Peterson
Subject: Re: Equiable Shapes

Hi, Natalie.

I don't know how much you were told about the assignment (what shapes 
to try, and so on), or what the expected answers are; but I'll suggest 
how I would look at the triangle. For other ideas, here are a couple 
of entries from our Dr. Math archives:

   Determining Equable Shapes
   http://mathforum.org/dr.math/problems/jonathan.6.15.99.html   

   Equable Shapes
   http://mathforum.org/dr.math/problems/laura.4.26.01.html   

The trouble with a triangle is that it depends on three variables 
(say, the lengths of the sides) rather than just one, as a square 
does. Worse, it is very hard to compute the area and the perimeter 
from the same variables. There is a formula to find the area from the 
sides, but it's pretty complicated. (See our FAQ on Formulas.) If, on 
the other hand, you know the base and altitude, you can find the area, 
but you need a third number to find the perimeter, and then it will 
involve square roots again.

I would suggest that you start, at least, by considering special 
triangles. Try an equilateral triangle first, which needs only the 
side length, so it will work much like the square. Then try a right 
triangle, which will be determined by the legs alone; you will find a 
relationship between them (rather than a single number) that makes the 
triangle "equable." An equilateral triangle will be similar.

Another approach would be to choose any triangle, find its area and 
perimeter, and then determine what to multiply it by to make a similar 
triangle that is "equable." That method might work well for the 
general case, and will confirm your results for the special cases.

At any rate, I would recommend working with easier shapes before you 
get to the triangle. Try some of these same ideas on circles and 
rectangles, and regular polygons (of which the square and equilateral 
triangle will be special cases). An interesting way to handle the 
rectangle is to think of one dimension, say the width, as being fixed, 
and then find what length is required to make it "equable." Then you 
can try the similar rectangle method, by choosing the ratio of length 
to width and finding the length and width to use.

Finally, I would have some fun with this by trying to find a common 
principle that lies behind "equability": is there some dimension or 
ratio that "equable" shapes tend to have in common (approximately), 
and any reason why that might be true?

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