Solution Hint: Algorthimic Problem 1st (How to find the shared area of two intersected circle)

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If you will go for the problem statement then you will find that you have to find the Area covered by intersection of two or three circles and the coordinates and radius is given.




Lets see how to find the shared area between intersection of two circles.

Let two circles are given.

1. center(x0,y0), radius (r0)
 2. center(x1,y1), radius (r1)

Let A be the centre of the circle ( x0, y0 ) and B be the centre of the other circle ( x1,y1 ).

Draw the circles with appropriate radii r0 and r1 so that there is a reasonable amount of overlap.  The length AB is calculated from the coordinates of the centre:

AB = sqrt{(x1-x0)^2 + (y1-y0)^2}

For convenience let this length be denoted by c.

The two circles intersect in two points which I will label C and D.

Now we must calculate the angles CAD and CBD, and we do this using the cosine formula. In fact it is half of these angles that we first calculate, using triangle CAB.

r0^2 = r1^2 + c^2 - 2*r1*c*cos(CBA)
cos(CBA) = (r1^2 + c^2 - r0^2)/(2*r1*c)

found CBA, then CBD = 2(CBA).

cos(CAB) = (r0^2 + c^2 - r1^2)/(2*r0*c)
and then
CAD = 2(CAB)

Express CBD and CAD in radian measure. Then we find the segment of each of the circles cut off by the chord CD, by taking the area of the sector of the circle BCD and subtracting the area of triangle BCD.
Similarly we find the area of the sector ACD and subtract the area of triangle ACD.

Area = ( 1/2 )( CBD ) r1 ^ 2 - ( 1/2 ) r1 ^ 2 * sin( CBD )
  + ( 1/2 )( CAD ) r0 ^ 2 - ( 1/2 ) r0 ^ 2 * sin( CAD )

Remember that for the area of the sectors you must have CBD and CAD in radians.

One more thing if the two circles are of the SAME radius please note that the area is symmetrical about the chord CD. Therefore, you only need to find the area in one half of the intersection and multiply by 2.

A shorter equation is

Area = 2 * ( ( 1/2 ) ( CBD ) r1 ^ 2 - ( 1/2 ) r1 ^ 2.sin( CBD ) ).

One more derivation if the redis is equal then you can find out using, this formula
Area = r^2*(q - sin(q))  where q = 2*acos(c/2r),
where c = distance between centers and r is the common radius.

I think now you can solve this. If something goes wrong, then post in the comments.

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1 comments On Solution Hint: Algorthimic Problem 1st (How to find the shared area of two intersected circle)

  • hmmmm nice……one other way is there..Its quite similar bt may be easy to understand.

    r0,r1 & c is known….so u can easly find area of ABCD..let it ‘A’.

    By cosine formula…find angle(CAD) and angle(CAB) then Find Ar(sector CAD) and Ar(sector CAB).Let it be ‘d’ and ‘b’ respectively.

    Now required area is= “d + b – A”

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