Diagonals in a Polygon
diagonal is a line segment that connects non-adjacent vertices in a polygon. Consider
the number of diagonals in a triangle, quadrilateral, pentagon, hexagon,
heptagon, and octagon. What pattern do you notice? Use this pattern to predict
the number of diagonals in a dodecagon.
First, create three segments to form a triangle. Since you are looking for diagonals you
should connect a segment vertex to vertex but the segments cannot be
adjacent. Therefore, the triangle
has no diagonal line segments.
create a quadrilateral. Connect each non-adjacent vertex with a line segment. You should see that there are only two
possible diagonal segments.
construct a pentagon. Again,
connect all non-adjacent vertices until all possible diagonal segments are
constructed. You should find that
a pentagonal figure should have 5 diagonal segments.
a hexagonal figure. Connect all
non-adjacent vertices until all possible diagonals have been created. When finished you should have 9
a heptagon. Connect all vertices
until all non-adjacent possible diagonals have been connected. You should have connected 14 different
an Octagon. Connect all vertices
until all non-adjacent possible diagonals have been connected. You should have connected 20 different
continued to count each of the polygon’s diagonals, I decided to make a
chart, listing the number of vertices and the number of diagonals
possible. I noticed that for each
polygon there were a different number of diagonals possible for each
vertex. The more sides the figure
had the more possible diagonals from each point. I found that each time you added a side you could also add
the additional diagonal from the vertex.
Example, a quadrilateral had 1 diagonal from each vertex, a pentagon had
2 diagonals from each vertex, a hexagon had 3 diagonals from each vertex,
etc. The pattern I used to
determine how many diagonals a particular figure could have was that I
multiplied the number of vertices by the number of diagonals from each vertex
and then halved it. Example, a
hexagon has 6 vertices with 3 diagonals coming from each, so
3=18, 18/2= 9, the hexagon should have 9 diagonals possible. It is not 18 because you cannot count
the same diagonal twice.
further exploration, I created the following formula:
the number of sides
a dodecagon has 54 possible diagonals.
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