Circle Overlap

Date: 11/08/2001 at 21:52:11
From: Justin Zaborowski
Subject: Circle Overlap

Both Circle A and Circle B have a radius of 1 unit. The centers of 
the two circles are 1 unit apart as well. Find the area of the union 
of the two circles.

I really am thankful for all of your help.

Date: 11/09/2001 at 09:01:42
From: Doctor Mitteldorf
Subject: Re: Circle Overlap

Dear Justin,

Here's a way to think about it.  

First, draw two unit equilateral triangles inside the overlapping 
area, with their vertices at the centers of the circles and the 
intersections of the circles. 

   

Can you see why these triangles must be equilateral?

Next, use Pythagoras to find the area of each equilateral triangle. 
You should get an answer of sqrt(3)/4 for each one.

Last, there are four lens-shaped pieces outside the equilateral 
triangles. Think of each one of these as the difference between (1/6 
of a unit circle) and (a unit equilateral triangle).  In other words, 
each lens should have an area of (pi*r^2)/6 - sqrt(3)/4.

Finally, add up the pieces: two equilateral triangles and 4 lens-
shaped differences.

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