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Scale Factor of Similar Shapes


Date: 05/12/2000 at 12:00:32
From: Bobi
Subject: Geometry

Two regular octagons have sides of lengths 21 and 28, respectively. 
Find:

   A) the scale factor
   B) the ratio of the perimeters
   C) the ratio of the areas of the octagons

I have no clue on how to do this at all. I haven't been able to figure 
out any of this.

Thank you!
Bobi


Date: 05/12/2000 at 13:01:12
From: Doctor Peterson
Subject: Re: Geometry

Hi, Bobi.

Do you know what a scale factor is? If you do, the first part is 
simple: it's just the ratio of the two numbers you were given.

For the others, you can actually ignore what the shape is; you don't 
have to figure out the perimeter or area of the octagon. If two 
figures are similar (the same shape), as all regular octagons are, 
then the ratio of any length in the figure (such as a diameter, or the 
perimeter) will be the same as the scale factor, and the ratio of 
areas will be the SQUARE of the scale factor.

You can see this more easily if you look at a square rather than an 
octagon. If the side of one square is S and the other is kS, the scale 
factor will be kS/S = k. The perimeters will be 4S and 4kS, and again 
the ratio is (4kS)/(4S) = k. The areas will be S^2 and (kS)^2 = k^2 
S^2; the ratio of these is (k^2 S^2)/S^2) = k^2, the square of the 
scale factor. Do you see how the scale factor gets squared? When you 
find the area, you are multiplying two multiples of k together, so the 
scale factor appears twice in the answer. Perhaps you can guess that 
the ratio of volumes of similar solid figures is the cube of the scale 
factor.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Geometry
Middle School Ratio and Proportion
Middle School Triangles and Other Polygons

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