Regions of a Circle

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Regions of a Circle

• By Alanna Roach and Amy Ziems

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Investigation

• If nodes are placed around the circumference of a
  circle, what is the maximum number of regions
  that can be formed?

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Nodes The points that are either equally or
unequally spaced around the circumference of a
circle. Regions The areas that are formed after
joining up all the nodes.
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Our hypotheses

• The more nodes placed around the circumference,
  the more important the spacing of the nodes are.
• When the nodes are equally placed as opposed to
  when they are not, there are more regions formed.
  
• Different shapes will be formed depending on the
  number of nodes and where they are placed.

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• The following slides show what can be produced
  when you place a certain amount of nodes around
  the circumference of a circle.

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One node equals one region. There are no
lines produced
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Two nodes equals two regions. There is one
line
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Depending on where the nodes are placed, the
length of the line increases when the nodes are
further apart, and decreases when the nodes are
closer together. The shape that is formed is
altered according to where the nodes are placed.

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Three nodes equals four regions. There are 3
lines
In this example, there is one triangle formed and
three semi-circles.
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Four nodes gives 4 semi-circles and four
triangles.
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Five nodes equals sixteen regions
As you can see there are many different shapes
formed and the equally spaced nodes form an
obvious star, from
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Six nodes equally placed equals thirty regions.
There are 15 lines
On all the diagrams, there is a semi-oval shape
formed when the nodes next to each other are
joined.
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Six nodes not equally spaced equals thirty
regions. There are 15 lines
After testing the different amounts of nodes on
circles, we found that having six nodes equally
spaced and one unequally spaced came out that the
regions were the same amount.
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Seven equally placed nodes equals fifty seven
regions. As the nodes increase the amount of
regions increases but they decrease in
size. There are twenty-one lines.
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3
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The line size varies, depending how far away the
node is it has to connect with. For example, the
red line (1-2) is much longer than the blue (1-3).
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Seven un equally spaced nodes equals fifty
regions. There are twenty-one lines
As you can see, the shapes produced by the lines
intersecting near the nodes, they are all
triangles, whilst the shapes in the middle of the
circle are varied shapes triangles,
quadrilaterals and pentagonal shapes.
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We found that after seven nodes the spacing
becomes more important, as when it is unequally
spaced there are less regions, and when they are
equally spaced there are more regions.
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As the number of nodes increase we tried to look
for a pattern between the number of nodes and the
shapes formed. We found that there was a positive
strong correlation between the number of nodes
and the number of semi-circles formed. There were
always the same amount each time. The red number
6 indicated that the nodes were placed unequally
and the shape that was affected was the
quadrilateral.
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This graph represents the relationship between
the number of nodes and the semi-circles formed.
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• As the lines intersect and cut into each other,
  they form new shapes, making more regions in the
  circle. This gives us the maximum number of
  regions in a circle.

2 regions when another line is added when
another line it gives 4 regions
intersects, it gives 7 regions
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When 5 or more nodes are joined together a star
is formed and the tips of these stars form
triangles. We investigated the amount of
triangles in each point of the star.
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We found that each time the number of nodes
increased by one the number of triangles in all
the stars. The pattern that we found was that
five was 5x1. Six was 6x2. Seven was 7x3. Eight
was 8x4. This proved that each was timed one
extra.
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• CONCLUSION
• When seven or more nodes are placed on a circle
  the amount of regions are different.