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### Trisecting a Circle with Parallel Cuts

```Date: 06/12/2002 at 09:08:18
From: Anonymous
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.

```

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

Hi,

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/
```
Associated Topics:
High School Conic Sections/Circles

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