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.