Overlapping Circles

Date: 07/14/99 at 22:38:13
From: Hastings
Subject: Geometry

Each of two overlapping circles has a radius of 6 inches. How long is 
the darkened portion of the circumferences of these circles, in 
inches?

     a) 48pi
     b) 24pi
     c) 20pi
     d) 18pi
     e) 16pi

The two circles overlap such that the perimeter of each circle touches 
the center of the other circle. The shaded region is the perimeter of
the figure formed when the circles are placed as described above.  

The answer is e) 16pi, but I don't know why. I tried to subtract the 
length of the circle's perimeter inside the figure from the total 
perimeter of the two circles. The total perimeter is 2(2xpix6) or 
24pi. Next I drew a line down the center of the figure, connecting the 
two points where the circles meet. Since the line is 180 degrees, or 
pi radians, and the radius from the origin (the point along this 
connecting line collinear to the centers) is 3, the length of the 
enclosed perimeter is 2(3pi), or 6pi.  24pi - 6pi = 18pi. I don't know 
what I'm doing wrong.

Date: 07/15/99 at 13:03:37
From: Doctor Peterson
Subject: Re: Geometry

Hi, Hastings.

You've done some good work, but you got sidetracked by a wrong 
assumption.

Here's your figure:

                *******        *******
            ****       **** ***       ****
         ***             ..*..            **
        *              .. /A\ .             **
       *              .  /   \ .              *
     **              .  /     \ ..             *
    *               .  /       \  .             *
    *               . /         \ .             *
    *               ./           \.             *
   *               ./             \.             *
   *             O +-------+-------+ P           *
   *               .\      X      /.             *
    *               .\           /.             *
    *               . \         / .             *
    *               .  \       /  .             *
     **              .  \     / ..             *
       *              .  \   / .              *
        *              .. \B/ .             **
         ***             ..*..            **
            ****       **** ***       ****
                *******        *******

It sounds as if you're imagining that O, A, P, and B are on a circle 
centered at X. They aren't. Since A, O, and B are already on a circle 
centered at P, they can't be the same distance from X as well.

What you do know is that the arc AOB is part of a circle, and you can 
figure out the central angle APB by looking at the triangles APO and 
OPB. Think carefully about the lengths of the sides of these 
triangles. Once you know the angle, you can find the arc length as a 
fraction of the circumference. Then fit that number into your original 
calculation and you'll be back on track.

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