Finding the Area of Overlap between Two Circles

Date: 04/16/2008 at 15:40:16
From: Mehrdad
Subject: A Formula

Suppose I have center coordinates of 2 circles and their radiuses.
Now how can I calculate the overlapped area of these 2 circles? 
(assume they have overlap)

Date: 04/18/2008 at 21:08:48
From: Doctor Ali
Subject: Re: A Formula

Hi Mehrdad!

Thanks for writing to Dr. Math.  One way to approach a problem like
yours is to use an integral.  I'll assume you are familiar with that
idea from calculus.

Let's assume that the radii are r1 and r2.  Do you accept that no
matter where these circles are in the plane, the answer is always the
same if the distance between the centers is the same?  If so, let's
assume that d is the distance between the centers.

Notice that you can evaluate d if you know the coordinates of the 
centers using the distance formula.  Assume that C1 and C2 are the
centers with these coordinates:

  C1(x1, y1)

  C2(x2, y2)

        ___________________________
  d = \/ (x2 - x1)^2 + (y2 - y1)^2

Now, I'll try to move the circles in a way that both centers are on 
the y-axis.  Take a look at this picture to visualize what I am saying
better:

   

We are looking for the area of the green region.  The two red points 
are the centers.  The distance between them is assumed to be d.  You 
can also see the x-axis and the y-axis in the picture.  No matter 
where they are, the area is always the same.

We want to write the equations of these two circles and find the area 
between them.  To simplify the equations, we will assume that one of 
the centers is on the origin.  This doesn't change anything with 
regard to the area.  Here's a picture showing it:

   

Remember that a full circle is not a function.  The upper semi-circle 
of the lower circle and the lower semi-circle of the upper circle is 
important here since those are the two semi-circles that actually
enclose the green area.

Recall that the general equation of a circle is

 (x - h)^2 + (y - k)^2 = r^2

where (h,k) are the coordinates of the center and r is the radius.

Thus, the equation of the lower circle will be:

  x^2 + y^2 = r1^2

Solving for y, the upper semi-circle of the lower circle will be:

  y = sqrt(r1^2 - x^2)

Do you know why this equation gives only the upper semi-circle?

The same goes for the upper circle with center at (0,d).  Full circle:

  x^2 + (y - d)^2 = r2^2

Lower semi-circle:

  y = d - sqrt(r2^2 - x^2)

Now you can find the intersection points for the two semi-circles.  
Set the equations equal, substitute your specific radii and value of
d, and solve for the two x-coordinates of the intersection points.

They will be the limits of the following integral where f(x)
represents the upper semi-circle of the lower circle and g(x)
represents the lower semi-circle of the upper circle:

  |  -                   |
  | |                    |
  | | [ f(x) - g(x) ] dx | = Area between f(x) and g(x)
  | |                    |
  |-                     |

Can you integrate the expressions?

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum
  http://mathforum.org/dr.math/