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Re: Area of two intersecting circles

  • To: mathgroup at smc.vnet.net
  • Subject: [mg123407] Re: Area of two intersecting circles
  • From: Ray Koopman <koopman at sfu.ca>
  • Date: Tue, 6 Dec 2011 03:12:12 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jbf968$l4j$1@smc.vnet.net>

On Dec 3, 11:58 pm, Scott Colwell <srcolw... at gmail.com> wrote:
> I have 2 disks named A and B.  They both have the same radius.
> Is there a function in mathematica that will find the area of
> the intersection between the two circles?

Let the disks have radius 1, and let their centers be at 0,0 and 0,c.
Then the area of their intersection is

In[1]:= Assuming[0 < c < 2, 4 Integrate[
          Boole[x^2 + y^2 < 1]*Boole[(x-c)^2 + y^2 < 1],
          {x, c-1, c/2}, {y, 0, 1}]]

Out[1]= (1/2)*((-c)*Sqrt[4 - c^2] + 4*ArcCos[c/2])

That simplifies (manually) to

2*( ArcCos[c/2] - (c/2)*Sqrt[1 - (c/2)^2] ),

If we take  c/2 = Cos[t/2]

then the area further simplifies to  t - Sin[t].

For the general case, use

c/2 = (center-to-center distance)/(diameter)

and multiply the area by r^2.



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