How to Increase the Sides to Double the Area of a Square

Date: 01/28/2005 at 12:16:10
From: colby
Subject: double the area of a square

I am puzzled as to how you would double the area of a square.  Do 
you multiply the sides by a decimal?  I can't find the right number to
multiply by, or the right solution.

Date: 01/30/2005 at 12:45:30
From: Doctor Wilko
Subject: Re: double the area of a square

Hi Colby,

Thanks for writing to Dr. Math!

Let's take an example.

Say you have a 2ft by 2ft square.  The area is 4 sq. ft.

   -------             
  |       |
  |4 ft^2 | 2 ft    (2 ft)^2 = 4 sq ft
  |       |
   -------             
      2 ft

Now say you want to double the area of this square to 8 sq. ft.

Your question is what do we have to multiply 2 by (on the original 
square) to be able to double the area (in this example, from 4 ft^2 
to 8 ft^2)?


   -------                        ----------  
  |       |                      |          |
  |4 ft^2 | 2 ft    ===>         | 8 ft^2   |  (2*x) ft
  |       |     (double area)    |          | 
   -------                       |          |
      2 ft                        ----------
                                      (2*x) ft


     2^2 = 4                      (2 * x)^2  =   8
                                   
                                   (2 * x)   =  sqrt(8)

                                    2 * x    =  2 * sqrt(2)   
 
                                        x    =  sqrt(2)

So it looks like in this example, you'd multiply 2 by sqrt(2), then 
square that to double the area.

Let's check it and see.

   (2 ft)^2 = 4 sq ft  (Area of original square)

If we multiply a side of length 2 by sqrt(2) and then square that, do 
we get 8 sq ft?

              (2 * sqrt(2))^2   =?  8 sq ft

  (2 * sqrt(2)) * (2 * sqrt(2)) =?  8 sq ft

  (2 * 2) * (sqrt(2) * sqrt(2)) =?  8 sq ft
     |               |
     |               |
     4       *       2           =   8 sq ft  (It checks!)


So, in general, to double the area of ANY square with a side of length 
s, multiply s by sqrt(2), then square that, i.e. [s * sqrt(2)]^2.

   ----------                        ---------------  
  |          |                      |               |
  | s^2 sq ft| s ft      ===>       |               |  s*sqrt(2) ft
  |          |       (double area)  | 2*(s^2) sq ft | 
   ----------                       |               |
      s ft                          |               |
                                     ---------------
                                      s*sqrt(2) ft

Does this help?  Please write back if you need anything else or are
confused about this.

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