The problem is to compute the area of the intersection of two circles of arbitrary radius.

In this post, the trivial cases (no intersection and one circle completely inside the other) are not discussed.

Let A be the center of the circle (x0,y0) of radius r0 and B be the center of the other circle (x1,y1) of radius r1.

We want to calculate the area of the green area in the figure 1.

*Figure 1: Area to compute*

Let’s add two points at the intersections between the two circles, C and D.

The area can therefore be decomposed in two sub-areas A1 and A2 being the left and the right parts of the intersection.

*Figure 2: Sub-areas A1 and A2*

The area of each sub-area can be calculated as the difference of the area of the pie and the triangle.

*Figure 3: How to calculate one sub-area*

First, and .

The area of the pie is proportionnal to the area of the whole circle

and therefore with alpha the angle of the pie you have the following relation:

We have to find the angles now.

so we just have to find BAC.

We know all the lengths of the edges of the triangle ABC and therefore

we can calculate its angles with the cosine rule (https://en.wikipedia.org/wiki/Law_of_cosines).

The length AB can be calculated simply from the coordinates of A and B

Similarly,

Now we juste have to calculate the area of the triangles and we are done.

Since we already have the length of two sides of the triangle and the angle in-between,

we can use that

Same for

The demonstration is easy:

Area of a triangle is

here

and so we finally get the expected result

*Figure 4: Area of the triangle DAC*

We finally have:

with

Links:

https://stackoverflow.com/questions/4247889/area-of-intersection-between-two-circles

http://mathforum.org/library/drmath/view/54785.html

http://mathworld.wolfram.com/Circle-CircleIntersection.html

This article is mostly inspeared of the explanation on mathforum.

The figures were done with https://www.geogebra.org