What is the area of overlapping region of two circles having radii r_1 and r_2 when the distance between their centres is c, given that r_1 +r


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Question Number 3877 by Rasheed Soomro last updated on 23/Dec/15

$W h a t i s t h e a r e a o f o v e r l a p p i n g$ $r e g i o n o f t w o c i r c l e s h a v i n g r a d i i$ $r_{1} a n d r_{2} w h e n t h e d i s t a n c e b e t w e e n$ $t h e i r c e n t r e s i s c, g i v e n t h a t r_{1} + r_{2} > c .$
Commented by prakash jain last updated on 24/Dec/15

$I will make the correction .$ $The above procedure is for centers on different$ $side of common chord .$
Commented by prakash jain last updated on 24/Dec/15

$Common chord AB$ $center O_{1} and O_{2}$ $O_{1} O_{2} cuts AB at X$ $∠ {AO}_{1} X = α$ $∠ {AO}_{2} X = β$ $AB = 2 r_{1} \sin α = 2 r_{2} \sin β$ $O_{1} X = r_{1} \cos α, {XO}_{2} = r_{2} \cos β$ $r_{1} \cos α + r_{2} \cos β = c$ $area of △ {AO}_{1} B = \frac{1}{2} AB \times O_{1} X = r_{1}^{2} \cos α \sin α$ $area of △ {AO}_{2} B = \frac{1}{2} AB \times O_{2} X = r_{2}^{2} \cos β \sin β$ $area of overlapin portion = area of$ $circle segment - area of triangle$ $Area of overlapping portion on side of$ $circle with radius r_{1} = α r_{1} - a r e a △ {AO}_{1} B$ $X = \frac{1}{2} α r_{1}^{2} - \frac{1}{2} r_{1}^{2} \sin 2 α . . . (1)$ $Area of overlapping portion on side of$ $circle with radius r_{2} = β r_{2} - a r e a △ {AO}_{2} B$ $Y = \frac{1}{2} β r_{2}^{2} - \frac{1}{2} r_{2}^{2} \sin 2 β . . . (2)$ $Next step is to find α, β$ $r_{1} \cos α + r_{2} \cos β = c \Rightarrow r_{1}^{2} \cos^{2} α = c^{2} - 2 {c r}_{2} \cos β + r_{2}^{2} \cos^{2} β$ $r_{1} \sin α = r_{2} \sin β \Rightarrow r_{1}^{2} \cos^{2} α = r_{1}^{2} - r_{2}^{2} + r_{2}^{2} \cos^{2} β$ $c^{2} - 2 {c r}_{2} \cos β + r_{2}^{2} \cos^{2} β = r_{1}^{2} - r_{2}^{2} + r_{2}^{2} \cos^{2} β$ $\cos β = \frac{c^{2} - r_{1}^{2} + r_{2}^{2}}{2 {c r}_{2}} . . . (3)$ $\cos α = \frac{c^{2} - r_{2}^{2} + r_{1}^{2}}{2 {c r}_{2}} . . . (4)$ $Total overlapping area is X + Y as given$ $in equation (1) and (2) . value of α and β$ $given in (3) and (4) .$
Commented by Yozzii last updated on 23/Dec/15

${I s n}^{'} t t h e a r e a o f a s e c t o r o f a c i r c l e o f$ $r a d i u s r p r o d u c e d b y a n g l e α g i v e n b y$ $\frac{1}{2} r^{2} α ? r α g i v e s a r c l e n g t h n o ?$ $I s t h e m e t h o d d i f f e r e n t i f b o t h$ $c e n t r e s a r e o n t h e s a m e s i d e o f t h e$ $c o m m o m c h o r d ?$
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