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Question Number 45366 by behi83417@gmail.com last updated on 12/Oct/18

Commented by behi83417@gmail.com last updated on 12/Oct/18

$s h a d e d a r e a = ?$
Commented by ajfour last updated on 12/Oct/18

	Answered by ajfour last updated on 12/Oct/18

	$x = \frac{r^{2}}{2} θ = \frac{9}{2} (180 ° - 20 °) \times \frac{π}{180 °}$ $= \frac{9}{2} \times \frac{8 π}{9} = 4 π$ $(s i n c e ∠ A C B = 150 ° - 130 ° = 20 °) .$
	Commented by behi83417@gmail.com last updated on 12/Oct/18

	$t h a n k s a l o t s i r A j f o u r .$
	Answered by MrW3 last updated on 12/Oct/18
	$=For any three circles= Radius of circle: black=r_1 green=r_2 red=r_2 (r_1 −r_2 )^2 =(r_1 −r_3 )^2 +(r_2 +r_3 )^2 −2(r_1 −r_3 )(r_2 +r_3 ) cos θ cos θ=(((r_1 −r_3 )^2 +(r_2 +r_3 )^2 −(r_1 −r_2 )^2 )/(2(r_1 −r_3 )(r_2 +r_3 ))) cos θ=((r_3 ^2 −r_1 r_3 +r_2 r_3 +r_1 r_2 )/((r_1 −r_3 )(r_2 +r_3 ))) cos θ=((r_2 (r_1 +r_3 )−r_3 (r_1 −r_3 ))/((r_1 −r_3 )(r_2 +r_3 ))) θ=cos^(−1) {((r_2 (r_1 +r_3 )−r_3 (r_1 −r_3 ))/((r_1 −r_3 )(r_2 +r_3 )))} A_(shade) =((r_3 ^2 (π−θ))/2)=(r_3 ^2 /2)[π−cos^(−1) {((r_2 (r_1 +r_3 )−r_3 (r_1 −r_3 ))/((r_1 −r_3 )(r_2 +r_3 )))}]$
	$= F o r a n y t h r e e c i r c l e s =$ $R a d i u s o f c i r c l e :$ $b l a c k = r_{1}$ $g r e e n = r_{2}$ $r e d = r_{2}$ ${(r_{1} - r_{2})}^{2} = {(r_{1} - r_{3})}^{2} + {(r_{2} + r_{3})}^{2} - 2 (r_{1} - r_{3}) (r_{2} + r_{3}) \cos θ$ $\cos θ = \frac{{(r_{1} - r_{3})}^{2} + {(r_{2} + r_{3})}^{2} - {(r_{1} - r_{2})}^{2}}{2 (r_{1} - r_{3}) (r_{2} + r_{3})}$ $\cos θ = \frac{r_{3}^{2} - r_{1} r_{3} + r_{2} r_{3} + r_{1} r_{2}}{(r_{1} - r_{3}) (r_{2} + r_{3})}$ $\cos θ = \frac{r_{2} (r_{1} + r_{3}) - r_{3} (r_{1} - r_{3})}{(r_{1} - r_{3}) (r_{2} + r_{3})}$ $θ = \cos^{- 1} {\frac{r_{2} (r_{1} + r_{3}) - r_{3} (r_{1} - r_{3})}{(r_{1} - r_{3}) (r_{2} + r_{3})}}$ $A_{s h a d e} = \frac{r_{3}^{2} (π - θ)}{2} = \frac{r_{3}^{2}}{2} [π - \cos^{- 1} {\frac{r_{2} (r_{1} + r_{3}) - r_{3} (r_{1} - r_{3})}{(r_{1} - r_{3}) (r_{2} + r_{3})}}]$
	Commented by behi83417@gmail.com last updated on 12/Oct/18

	$t h a n k y o u s o m u c h s i r .$ $h o w c a n w e f i n d r_{1} a n d r_{2} ?$
	Commented by MrW3 last updated on 12/Oct/18

	Commented by MrW3 last updated on 12/Oct/18

	$t h i s s h o w s t h e q u e s t i o n I s o l v e d . i t$ $i s n o t d i r e c t l y y o u r q u e s t i o n .$
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