My Math Forum area of the green section

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October 30th, 2012, 06:05 AM   #1
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area of the green section

eight squares (side length 1) put together, please find the area of the green section
[attachment=0:gsh3zdan]area of green section .jpg[/attachment:gsh3zdan]
$Ans :\,\,\,\frac {2\pi}{3}$
Attached Images
 area of green section .jpg (12.4 KB, 115 views)

October 30th, 2012, 08:17 AM   #2
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Re: area of the green section

Hello, Albert.Teng!

Quote:
 Eight squares (side length 1) put together. Find the area of the shaded region. Answer: $\frac{2\pi}{3}$ Code:  * * * A * - - - - - * B *:::::::::::::::* C *- - - - - - - - -* D * * E *---------*---------* F : - -2- - O - -2- - :

Draw $OA,\,OB,\,OC,\,OD.$
Note that they all have length 2.

$\text{W\!e find }\text{that: }\,\angle AOB\,=\,60^o,\:\angle COD \,=\,120^o.$

$\text{Area of sector }CDO:\;\frac{1}{3}\pi (2^2) \,=\,\frac{4}{3}\pi$

$\text{Area of sector }AOB:\;\frac{1}{6}\pi (2^2) \,=\,\frac{2}{3}\pi$

$\text{Area of }\Delta AOB:\;\frac{\sqrt{3}}{4}(2^2) \,=\,\sqrt{3}$

$\text{Area of}\text{ segment }AB:\;\frac{2\pi}{3}\,-\,\sqrt{3}$

$\text{Area of }\Delta COD:\:\frac{1}{2}(2^2)\sin120^o \,=\,\sqrt{3}$

$\text{Shaded area} \:=\:\text{(sector }CDO)\,-\,\(\Delta COD) \,-\,\text{(segment } AB)$

[color=beige]. . . . . . . . . . . [/color]$=\;\frac{4\pi}{3}\,-\,\sqrt{3} \,-\,\left(\frac{2\pi}{3}\,-\,\sqrt{3}\right)$

[color=beige]. . . . . . . . . . . [/color]$=\;\frac{2\pi}{3}$

I see that LaTeX is still misspelling my words.
So I have to take heroic measures to overcome it.

 October 30th, 2012, 10:47 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,095 Thanks: 1905 What difficulty occurred with LaTeX? It seemed to work for me.
October 30th, 2012, 09:04 PM   #4
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Re: area of the green section

the area of the green section is the same as the area of a sector with radius 2 and central angle :$\frac {\pi}{3}$

so the area is : $\frac {2\pi}{3}$
[attachment=0:ke4xayyo]area of green section~.jpg[/attachment:ke4xayyo]
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 area of green section~.jpg (21.5 KB, 69 views)

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