# Area of rectangle with embedded parallelograms

• Jan 20th 2010, 08:43 AM
ncooks
Area of rectangle with embedded parallelograms
Hello - I am new to the forum, and am at a loss at how to solve this problem. I need to find the area of the shaded region. I created a picture in paint. Can I get some help with this, and an explanation on how to solve this?

Thank youQ\!
• Jan 20th 2010, 09:58 AM
Soroban
Hello, ncooks!

Welcome aboard!

Quote:

Find the area of the shaded region.
Code:

              9       * - - - - - - - *       |:::::::::::::::*       |:::::::::::*  | 2       |:::::::*    ..*       |:::*      *:::*       *      *:::::::|       |  *:::::::::::*   13 *::::::::::*    | 2       |:::::::*    ..|       |:::*      *:::*       *      *:::::::|       |  *:::::::::::|       *:::::::::::::::|       |:::::::::::::::|       * - - - - - - - *

The area of the rectangle is: .$\displaystyle 9 \times 13 \:=\:117\text{ cm}^2$

Assuming that the diagonal lines are parallel,
. . the white regions are two congeuent parallelograms,
. . each with base 2 and height 9.

The white region has area: .$\displaystyle 2 \times (2\times 9) \:=\:36\text{ cm}^2$

Therefore, the area of the shaded region is: .$\displaystyle 117 - 36 \:=\:81\text{ cm}^2$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An easier way . . .

Remove the white strips and push the shaded regions together.
Code:

              9       * - - - - - - - *       |:::::::::::::::*       |:::::::::::*:::|       |:::::::*:::::::|     9 |:::*:::::::::::*       *:::::::::::*:::|       |:::::::*:::::::|       |:::*:::::::::::|       * - - - - - - - *