Converting to Square Units

Date: 01/07/2002 at 09:11:56
From: Joey
Subject: Converting to kilometers

How do I convert the number 47,224 sq miles to kilometers?

Date: 01/07/2002 at 10:08:21
From: Doctor Ian
Subject: Re: Converting to kilometers

Hi Joey,

Strictly speaking, you can't. Square miles are a measure of area, and 
kilometers are a measure of length.  

Do you mean, how can you convert square miles to square kilometers?  
Let me show you how to do it with different units, and you can apply 
the same reasoning to your own problem. 

Suppose I have something that is 4 square inches in area:

     1 in      1 in
  +---------+--------+
  |         |        |
  |         |        | 1 in
  |         |        |
  +---------+--------+
  |         |        |
  |         |        | 1 in
  |         |        |
  +---------+--------+

Now, I know that there are 2.54 centimeters in an inch. So I can label 
my picture this way:

    2.54 cm   2.54 cm
  +---------+--------+
  |         |        |
  |         |        | 2.54 cm
  |         |        |
  +---------+--------+
  |         |        |
  |         |        | 2.54 cm
  |         |        |
  +---------+--------+

So the area is cm is 

  (2 * 2.54) * (2 * 2.54)

If we look a little more closely, we can see what is going on:

  4 in^2 = (2 in) * (2 in)

                   2.54 cm            2.54 cm
         = (2 in * -------) * (2 in * -------)
                   1 in               1 in

                              2.54 cm
         = (2 in) * (2 in) * (--------)^2
                                1 in

          
                     2.54 cm
         = 4 in^2 * (--------)^2
                     1 in

So whereas for length, I would multiply by a conversion factor, e.g., 

                35 ziglofs
  21 blinches * ----------- = 21 * (35/17) ziglofs
                17 blinches

for area I have to multiply by the square of the conversion factor, 
e.g., 

                   35 ziglofs
  21 blinches^2 * (-----------)^2 = 21 * (35/17)^2 ziglofs^2
                   17 blinches

Does that make sense?  (Can you guess what you have to do for volume?)

There are 1.609 kilometers in a mile. Can you take it from here?

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