# Get the size of an area defined by 2 overlapping circles

I have two circles, like this:

I know the radii of the circles, and I know the X + Y of the center of both circles.

Can I get the size of the area that is black in my picture?

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–  lhf Jan 10 '12 at 11:37
I've tried asking a Math and a Physics teacher, they both didn't know ^^ –  Lolmewn Jan 10 '12 at 11:48
Google finds this: mathworld.wolfram.com/Circle-CircleIntersection.html –  lhf Jan 10 '12 at 11:52
Oh my, that looks crazy complicated. Isn't there an easier way to do this? –  Lolmewn Jan 10 '12 at 11:54
Do you mean easier to understand or easier to implement? Because equation 14 is pretty easy to implement. Just code up the expression and don't think about it too much. :) (And for what it's worth, I wouldn't hold out hope for a simpler formula. Areas of intersections tend to have complicated closed forms.) –  Rahul Jan 10 '12 at 12:12

A formula for the area is worked out in Circle-Circle Intersection at Wolfram MathWorld.

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This link helped me out in this situation. The MathWorld version is very descriptive, but I think the above link explains what is going on more clearly. I created a javascript function for this purpose, given below in case it is useful to anyone else.

function areaOfIntersection(x0, y0, r0, x1, y1, r1)
{
var rr0 = r0*r0;
var rr1 = r1*r1;
var c = Math.sqrt((x1-x0)*(x1-x0) + (y1-y0)*(y1-y0));
var phi = (Math.acos((rr0+(c*c)-rr1) / (2*r0*c)))*2;
var theta = (Math.acos((rr1+(c*c)-rr0) / (2*r1*c)))*2;
var area1 = 0.5*theta*rr1 - 0.5*rr1*Math.sin(theta);
var area2 = 0.5*phi*rr0 - 0.5*rr0*Math.sin(phi);
return area1 + area2;
}

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