# find area of shaded area (yellow)

• September 2nd 2007, 08:23 AM
bells
find area of shaded area (yellow)
2 quadrants within a square of length 10cm.

how to find the area of the shaded area in yellow?

thanks!

http://img.photobucket.com/albums/v3...untitled-7.jpg
• September 2nd 2007, 08:29 AM
Krizalid
Find an equilateral triangle. There's a lot of hidden information there.
• September 2nd 2007, 08:52 AM
red_dog
You can split the area in two sectors with the same area and an equilateral triangle.
Each sector has an angle of $30^{\circ}$.
• September 2nd 2007, 09:25 AM
Soroban
Hello, bells!

Did you catch red_dog's excellent hint?

Quote:

Two quadrants within a square of length 10cm.
. . Find the area of the shaded region.

http://img.photobucket.com/albums/v3...untitled-7.jpg

Label the square $A\!-\!B\!-\!C\!-\!D$, starting at the upper-left and labelling clockwise.
Label the intersection of the two arcs with $O$.
Draw radii $OC$ and $OD$.

Triangle $OCD$ is equilateral with side 10cm.
. . I'm sure you can find its area.

Sectors $DAO$ and $CBO$ have radius 10 and central angles 30°.
. . and you can find their areas, too, right?

• September 2nd 2007, 03:06 PM
bells
why is it an equilateral triangle?

thanks.
• September 2nd 2007, 09:37 PM
earboth
Quote:

Originally Posted by bells
why is it an equilateral triangle?

thanks.

Hello,

because all three sides have the same length.

I used
- red_dog's suggestion
- Sorobans labeling

see attachment
• September 2nd 2007, 09:49 PM
Jhevon
Quote:

Originally Posted by bells
why is it an equilateral triangle?

thanks.

earboth gave you detailed information on how to find the answer, but allow me to fill in a bit of the background.

recall that the yellow regions are outlined by circular arcs with radius 10. consider the left arc. any line drawn from the bottom left corner to any point along the arc will have the length of the radius, which is 10. the same thing holds true for the arc on the right. since the base of the triangle drawn is one side of the square, it will have a length of 10 as well. thus we have a triangle with sides 10 units of length each, thus it is an equilateral triangle
• September 4th 2007, 06:11 AM
janvdl
Wow, this was an excellent problem. I like it. :)
• September 4th 2007, 08:34 AM
bells
got it!

thanks! :)