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 November 13th, 2010, 11:01 PM #1 Newbie   Joined: Nov 2010 Posts: 4 Thanks: 0 Help me! Find the area Find the area of the crescent-shaped region(called a lune) bounded by arcs of circles with radii r and R(see the figure) http://www.mathhelpforum.com/math-help/ ... titled.png Thank you very much
November 14th, 2010, 03:00 AM   #2
Senior Member

Joined: Oct 2010
From: Changchun, China

Posts: 492
Thanks: 14

Re: Help me! Find the area

clue:
"area of pink" + "area of green"= 'half area of the smaller circle", so you need to get the area of pink.

[attachment=0:1f44t6p0]Untitled.PNG[/attachment:1f44t6p0]
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 November 14th, 2010, 03:46 AM #3 Newbie   Joined: Nov 2010 Posts: 4 Thanks: 0 Re: Help me! Find the area Can you help me do it?
November 14th, 2010, 03:47 AM   #4
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Joined: Nov 2010

Posts: 4
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Re: Help me! Find the area

Quote:
 Originally Posted by stainburg clue: "area of pink" + "area of green"= 'half area of the smaller circle", so you need to get the area of pink. [attachment=0:2bj8v9te]Untitled.PNG[/attachment:2bj8v9te]
Can you help me to do it?
Thank you very much.

 November 14th, 2010, 05:48 AM #5 Senior Member     Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14 Re: Help me! Find the area $S_a=\frac{\pi{r^2}}{2}-S_b$ $S_b=S_{\text{sector}}-S_{\bigtriangleup}$ $S_{\text{sector}}=\frac{1}{2}\alpha{R^2}\longright arrow^{{\alpha=2\arcsin{\frac{r}{R}}}}R^2\arcsin{\ frac{r}{R}}$ $S_{\bigtriangleup}=\frac{1}{2}\times2r\times\sqrt{ R^2-r^2}=r\sqrt{R^2-r^2}$ $S_b=S_{\text{sector}}-S_{\bigtriangleup}=R^2\arcsin{\frac{r}{R}}-r\sqrt{R^2-r^2}$ $S_a=\frac{\pi{r^2}}{2}-S_b=\frac{\pi{r^2}}{2}-(R^2\arcsin{\frac{r}{R}}-r\sqrt{R^2-r^2})=\frac{\pi{r^2}}{2}+r\sqrt{R^2-r^2}-R^2\arcsin{\frac{r}{R}}$

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# the crescent shaped region bounded by two

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