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### 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

- Doctor Wilko, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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