Trisecting a Circle with Parallel Cuts

Date: 06/12/2002 at 09:08:18
From: Kim Albrektsson
Subject: Geometry

Hi,

I wonder if it's possible to draw two parallel lines in a circle so 
that you get three pieces that have the same area.

Grateful for a quick answer.

Date: 06/12/2002 at 09:43:38
From: Doctor Ian
Subject: Re: Geometry

Hi Kim,

Take a look at our FAQ on segments of circles:

    http://mathforum.org/dr.math/faq/faq.circle.segment.html 

Pay particular attention to case 13, in which you have the radius 
of the circle (r) and the angle that subtends the arc of the 
segment (theta). 

The area of the segment is 

      theta - sin(theta)
  r^2 ------------------
             2

We would like for the area of the segment to be 1/3 of the area 
of the entire circle, since then we could cut another segment 
just like it off the other side of the circle, and end up with 
three equal-sized pieces. 

So let's set the area of the segment equal to 1/3 the area of the 
entire circle:

      theta - sin(theta)   pi r^2
  r^2 ------------------ = ------
             2                3

We can cancel r^2 from both sides to get 

      theta - sin(theta)   pi 
      ------------------ = --
             2              3

and multiply both sides by 2 to get 

                           2 pi
      theta - sin(theta) = ----  = 2.09
                             3

Note that 2pi/3 radians is 120 degrees, so theta is going to be 
larger than 120 degrees.  How much larger?  Could it be 150 
degrees? 

  5pi/6 - sin(5pi/6) = 2.62 - 0.5

                     = 2.12

which is pretty close. You could try angles a little smaller than 
150 degrees if you want to get more precision. 

(This kind of equation needs to be solved in the same way as a 
square root, namely by guessing the answer and checking it.) 

But when you find the angle theta to whatever precision makes you 
happy, you can cut the pizza, by drawing two diagonals across the 
pizza, 

                *   *
            *           *

         A                 B

        *         C         *

         B'                A'

            *           *
                *   *
             
such that one of the angles between AA' and BB' is theta.  Then two 
parallel cuts, from A to B and from B' to A', will cut the pizza 
into thirds. 
             
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/