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### Proportions of Exact Enlargements

Date: 03/18/98 at 03:57:18
From: Emma J.
Subject: Ratio of 2 cylinders

A two-liter can is an exact enlargement of a one-liter can. Explain
why their base radii are in the ratio 1:(the cube root of 2).

I have no idea how to explain this conclusion and would be grateful
for some help.

Thanks.

Date: 03/18/98 at 09:56:28
From: Doctor Sam
Subject: Re: Ratio of 2 cylinders

Emma,

"Exact enlargement" is an informal way of saying that the two cans are
similar. Similar figures, whether they are 2D-plane figures or 3D-
shapes, are proportional. A general principle that can be proved with
a little geometry relates the volumes, areas, and corresponding
lengths of similar figures.

Here is the principle. I'll follow it up with a few examples to try to
make it meaningful.

In similar figures, corresponding lengths are proportional (that
is, have the same ratio).

In similar figures, corresponding areas are proportional to the
squares of corresponding lengths.

In similar figures, corresponding volumes are proportional to the
cubes of corresponding lengths.

For example, take a 3 x 5 rectangle. If you enlarge it with a
photocopier, you might get a rectangle that is 6 x 10 or 9 x 15.
Corresponding sides have the same ratio: 6/3 = 10/5; 9/3 = 15/5. The
areas of these rectangles are 15, 60, and 135 respectively. The ratio
of areas: 15/60 = (3/6)^2. Note that 15/60 = 1/4 and 3/6 = 1/2 and (1/
2)^2 = 1/4.

Another example: take a 2 x 2 x 2 cube. It is similar to a 5 x 5 x 5
cube.

The ratio of corresponding sides is 2/5.
The ratio of areas of corresponding faces is 4/25 = (2/5)^2.
The ratio of their volumes is 8/125 = (2/5)^3.

volumes is 2/1. Corresponding lengths (the radii of their bases or
their heights) will have ratio (a/b) and (a/b)^3 = 2/1. That should be
enough background to show the relationship that you want.

-Doctor Sam, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Ratio and Proportion
Middle School Two-Dimensional Geometry

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