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-2-gt-c-

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

\boldsymbol r_{1} a n d \boldsymbol 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 \boldsymbol c, g i v e n t h a t \boldsymbol r_{1} + \boldsymbol r_{2} > \boldsymbol 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 \dots (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 \dots (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}} \dots (3)

\cos α = \frac{c^{2} - r_{2}^{2} + r_{1}^{2}}{2 {c r}_{2}} \dots (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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