Miles Squared vs. Square Miles

Date: 07/09/99 at 07:17:45
From: Connor Wlodarczak
Subject: A question of area

How are these two different? I don't know!

     10 square miles
     10 miles square

I thought they were the same, but my friend says they are not.

Thank you very much,
Connor

Date: 07/09/99 at 08:50:00
From: Doctor Peterson
Subject: Re: A question of area

Hi, Connor. Good question.

I'm not sure whether there's a difference between "ten miles square" 
and "ten miles squared," but if this is a mile:

     +---+

then this is a "one mile square," a square one mile on each side whose 
area is called one square mile:

     +---+
     |   |
     +---+

and this is ten square miles:

     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+

But "ten miles square" (or "ten miles squared") means we make a square 
ten miles on a side:

     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+

If you count, or think a bit, you'll see that 10 miles square is 
actually 100 square miles, ten times as much as 10 square miles.

Words like "square" and "squared" can be tricky, which is why in math 
we use symbols rather than words, so we can make it clear exactly what 
we mean.

If you've seen how we write "squared numbers" in math, this can be 
written as
               2             2 
     (10 miles)  = 100 (miles )

     (10 miles) squared = 100 (square miles)

I hope this makes the difference clear to you.

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