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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
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 >c.
Whatistheareaofoverlappingregionoftwocircleshavingradiir1andr2whenthedistancebetweentheircentresisc,giventhatr1+r2>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.
Iwillmakethecorrection.Theaboveprocedureisforcentersondifferentsideofcommonchord.
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=2r_1 sin  α=2r_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=(1/2)AB×O_1 X=r_1 ^2 cos αsin α  area of △AO_2 B=(1/2)AB×O_2 X=r_2 ^2 cos βsin β  area of overlapin portion=area of  circle segment − area of triangle  Area of overlapping portion on side of  circle with radius r_1 =αr_1 −area △AO_1 B       X=(1/2)αr_1 ^2 −(1/2)r_1 ^2 sin 2α     ...(1)  Area of overlapping portion on side of  circle with radius r_2 =βr_2 −area △AO_2 B       Y=(1/2)βr_2 ^2 −(1/2)r_2 ^2 sin 2β      ...(2)  Next step is to find α, β   r_1 cos α+r_2 cos β=c⇒r_1 ^2 cos^2 α=c^2 −2cr_2 cos β+r_2 ^2 cos^2 β  r_1 sin  α=r_2 sin β⇒r_1 ^2 cos^2 α=r_1 ^2 −r_2 ^2 +r_2 ^2 cos^2 β  c^2 −2cr_2 cos β+r_2 ^2 cos^2 β=r_1 ^2 −r_2 ^2 +r_2 ^2 cos^2 β  cos β=((c^2 −r_1 ^2 +r_2 ^2 )/(2cr_2 ))          ...(3)  cos α=((c^2 −r_2 ^2 +r_1 ^2 )/(2cr_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).
CommonchordABcenterO1andO2O1O2cutsABatXAO1X=αAO2X=βAB=2r1sinα=2r2sinβO1X=r1cosα,XO2=r2cosβr1cosα+r2cosβ=careaofAO1B=12AB×O1X=r12cosαsinαareaofAO2B=12AB×O2X=r22cosβsinβareaofoverlapinportion=areaofcirclesegmentareaoftriangleAreaofoverlappingportiononsideofcirclewithradiusr1=αr1areaAO1BX=12αr1212r12sin2α(1)Areaofoverlappingportiononsideofcirclewithradiusr2=βr2areaAO2BY=12βr2212r22sin2β(2)Nextstepistofindα,βr1cosα+r2cosβ=cr12cos2α=c22cr2cosβ+r22cos2βr1sinα=r2sinβr12cos2α=r12r22+r22cos2βc22cr2cosβ+r22cos2β=r12r22+r22cos2βcosβ=c2r12+r222cr2(3)cosα=c2r22+r122cr2(4)TotaloverlappingareaisX+Yasgiveninequation(1)and(2).valueofαandβgivenin(3)and(4).
Commented by Yozzii last updated on 23/Dec/15
Isn′t the area of a sector of a circle of  radius r produced by angle α given by  (1/2)r^2 α?   rα gives arc length no?   Is the method different if both   centres are on the same side of the  commom chord?
Isnttheareaofasectorofacircleofradiusrproducedbyangleαgivenby12r2α?rαgivesarclengthno?Isthemethoddifferentifbothcentresareonthesamesideofthecommomchord?

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