Circle Inscribed in Sector

Date: 05/20/99 at 17:38:29
From: Chris Wipke
Subject: Geometry

The arc of a sector has degree measure 60 degrees. The radius of the 
sector is 12 inches. Find the area of the circle that can be inscribed 
in the sector. 

I have drawn a diagram of a sector whose arc measure is 60 degrees, 
and have constructed a line perpendicular to the lower segment forming 
the arc. I do not know what to do from here, or if I am right so far. 
Please provide me the way to figure out the problem, along with the 
correct answer.

Date: 05/21/99 at 13:04:25
From: Doctor Peterson
Subject: Re: Geometry

Hi, Chris. Interesting problem!

Here's how I understand your picture:

             **
            /  ****
           /  +++++**
          +++       +++
         +            *+
        +       +       +
       /+               +*
      /  +             + *
     /60  +++       +++   *
    *---------+++++-------*
               12

I would draw in a couple lines: a radius from the center of the 
inscribed circle to its point of tangency with a radius of the sector, 
and a radius of the sector passing through the center of the 
inscribed circle.

             A
             **
            /  ****
           /  +++++**  C
          +++       +++
         +      D  /r *+
        +       +       +
       /+    /  |       +*
      /  +/     |r     + *
     / /  +++   |   +++   *
    *---------+++++-------*
   O            E          B

Now because the circle is tangent to the arc, OC is a radius of the 
arc and has length 12. OD, the hypotenuse of a right triangle, has 
length 12-r. But triangle ODE is a 30-60-90 triangle, so we know this 
hypotenuse is exactly twice r. This will give you an equation you can 
solve for r.

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