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### Dividing a Circle using Six Lines

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Date: 08/29/2001 at 11:35:27
From: Jordy G
Subject: Geometry

What is the largest number of regions into which you can divide a
circle with six lines? Is there a formula? How can you get this
answer? I've tried drawing diagrams but I can't find a way to make
sure they are correct. I can only get 14, but I know there are more.
```

```
Date: 08/29/2001 at 14:29:41
From: Doctor Rob
Subject: Re: Geometry

Thanks for writing to Ask Dr. Math, Jordy.

There is a formula.  For n lines, you can get at most

(n^2+n+2)/2

regions.  When n = 6, this gives 22 regions.

To maximize the number of regions, draw your lines like this:

Start with one line. Label the two points where it meets the circle as
P1 and Q1.

Pick points P2 and Q2 such that they are encountered in the order
P1,P2,Q1,Q2,P1 going around the circle clockwise. Connect P2 and Q2 to
make the second line, and four regions.

Between P2 and Q1 pick point P3. Between Q2 and P1 pick point Q3.
Connect them with a line, but, if necessary, move Q3 a little to make
sure that the line does not pass through any of the intersections of
any previously drawn lines. That will give you seven regions.

Between P3 and Q1 pick point P4. Between Q3 and P1 pick point Q4.
Connect P4 to Q4 with a line, but, if necessary, move Q4 a little to
make sure that the line does not pass through any of the intersections
of any previously drawn lines. That will give you 11 regions.

Continue in this way until you have 6 lines and 22 regions.

P1-P6 and Q1-Q6 can be chosen to be 12 adjacent vertices of a regular
n-gon inscribed in the circle, if n > 12. It is easy to construct a
regular 16-gon by starting with a diameter of the circle and bisecting
the central angles until they are all equal to 360/16 = 22.5 degrees.
That will locate 16 equally-spaced points around the circumference of
the circle. Pick any 12 that are adjacent, and label them P1, P2, ...,
P6, Q1, ..., Q6. Then connect P1Q1, P2Q2, ..., P6Q6. If you have done
this carefully, you will be able to see and count the 22 regions.

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

```
Date: 08/29/2001 at 17:01:10
From: Doctor Greenie
Subject: Re: Geometry

Hi, Jordy -

There is a formula - as I see another doctor here has already shown
you.

But there is a way you can get the answer without the formula, by
logically analyzing what happens as you draw more and more lines.
And, if you know the correct mathematical techniques, this logical
analysis can lead you to a derivation of the formula.

I have always found mathematics to be more enjoyable when I can see
where a formula comes from, rather than having to regard it as some
sort of mathematical magic.

So here is how you can analyze this problem, at least far enough to

You are going to analyze how and when the number of regions increases
as you draw more lines through the circle. You will find that, if you
think of actually physically drawing each line, the number of regions
increases by one at a time; each increase is the result of an existing
region being cut into two separate regions.

Start with a circle; before you draw any line through the circle, it
is a single region.

Now, start drawing the first line through the circle, and consider at
what point(s) you add another region to the diagram. When you are
almost all the way from your starting point to the other side of the
circle, there is still only one region; it is not until you reach the
other side that you have divided an existing region (in this case, the
entire circle) into two parts. So, when you draw the first line
through the circle, there is one point at which one of the existing
regions becomes two regions: when you reach the opposite side of the
circle. So we have:

1st line drawn through circle ==> number of additional regions = 1

Next, start drawing the second line through the circle, and consider
at what point(s) you add another region to the diagram. Of course,
since you want to create the largest number of regions in the circle,
you want your second line to intersect the first line inside the
circle. So you start at some point on the circle, and when you reach
the first line somewhere inside the circle you have added another
region by dividing an existing region into two parts. And then when
you reach the other side of the circle you have again added another
region by dividing an existing region into two parts. So, when you
draw the second line through the circle, there are two points at
which one of the existing regions becomes two regions: when you reach
the first line you drew, and when you reach the opposite side of the
circle. So we have:

2nd line drawn through circle ==> number of additional regions = 2

Now, start drawing the third line through the circle, and consider at
what point(s) you add another region to the diagram. Of course, since
you want to create the largest number of regions in the circle, you
want this third line to intersect both of the first two lines at
different points inside the circle (if your third line intersects the
first two lines at their point of intersection, you don't get the
maximum possible number of regions). So you start at some point on the
circle, and when you reach the first line somewhere inside the circle
you have added another region by dividing an existing region into two
parts, and when you reach the second line you have again added another
region. And when you reach the other side of the circle you have again
added another region by dividing an existing region into two parts.
So, when you draw the third line through the circle, there are three
points at which one of the existing regions becomes two regions: when
you reach each of the the first two lines you drew, and when you reach
the opposite side of the circle. So we have:

3rd line drawn through circle ==> number of additional regions = 3

I think you can see where the analysis will go from here....

When you draw the n-th line, you create n new regions -- (n-1) new
regions, one for each time you intersect one of the previous (n-1)
lines that you drew, and a last time when you reach the other side of
the circle.

So you can answer your specific question by making a simple table

number of   number of   total number
lines    new regions   of regions
--------------------------------------
0            --            1
1             1            2
2             2            4
3             3            7
4             4           11
5             5           16
6             6           22

Then there are various mathematical techniques (which I won't go into
here) for examining the sequence of the number of regions

1, 2, 4, 7, 11, 16, 22, ...

to come up with the formula for the maximum number of regions formed
by n lines:

n^2+n+2
# regions with n lines = ---------
2

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

```
Date: 08/29/2001 at 21:13:47
From: Anonymous
Subject: Re: Geometry

Thank you so much!
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Sequences, Series

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