Polygons

Transcription

Polygons

Polygons
Polygons
When discussing polygons, some like
to refer to the parts, Poly and gon,
stating that this means “many sides”.
Many
??? Sides ???
Polygons
However, while “many” is fine for
Poly, is “sides” truly correct for gon?
What about “angles”?
Many
??? Angles ???
Polygons
It is best to check the word origin.
Many ??? Sides ???
Many ??? Angles ???
Polygon?
• From Latin: polygōnum
• From Greek: polýgōnon (noun use of
neuter of polýgōnos)
Meaning “many-angled”
What is Polygon?
What is Polygon?
• A closed shape formed by coplanar
line segments connected end to end
What is Polygon?
• A closed figure bounded by line
segments
What is Polygon?
segments
• A closed/bounded (geometric) shape
determined by line segments which
intersect in pairs at endpoints
What is Polygon?
segments
Polygon or Not a Polygon?
Let us check our definitions …
• A closed shape formed by coplanar line
segments connected end to end
• A closed figure bounded by line segments
Sides of Polygon
• The line segments
Angles of Polygon
• Formed by pairs of intersecting line
segments
Angles of Polygon
segments
• Formed by the sides of the polygon
Angles of Polygon
segments
• Formed by the sides of the polygon
• Also known as the vertex angles
Vertices of Polygon
• Points at which the endpoints of
line segments (in pairs) intersect
Vertices of Polygon
• Points at which the endpoints of
line segments (in pairs) intersect
• Points at which a pair of sides of a
polygon intersect
Adjacent Sides of Polygon
• Two sides of a polygon that have a
common vertex
Adjacent Vertices of Polygon
• Two vertices that are endpoints of
a side of a polygon
Vertices/Angles of Polygon
• The sides of polygons intersect only
at the endpoints of line segments
for adjacent sides.
• A pair of adjacent sides of a
polygon form an angle of the
polygon.
• The point at which a pair of
adjacent sides of a polygon
intersect is a vertex of the
polygon.
How many sides of a polygon
can intersect at any point?
• Recall: Two sides of a polygon
intersect at a common vertex.
• Recall: Two sides of a polygon
intersect at a common vertex.
2
Polygons
Not Polygons
Complex Polygons
• The sides of the polygon
cross/intersect one another*
Complex Polygons
• The sides of the polygon
cross/intersect one another*
*Some refer to these a self-intersection polygons while others do
not consider them to be polygons
 http://mathworld.wolfram.com/Polygon.html
 http://www.mathopenref.com/polygon.html
 http://mathworld.wolfram.com/SimplePolygon.html
Is this a Polygon???
Polygons
• Classified/named by the number of
sides/angles
Name/Type of Polygons
•
•
•
•
•
•
•
Triangle – 3 sides
Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon – 6 sides
Heptagon – 7 sides
Octagon – 8 sides
Nonagon (enneagon) – 9 sides
•
•
•
•
•
•
•
Decagon – 10 sides
Hendecagon (undecagon) – 11 sides
Dodecagon – 12 sides
tridecagon (triskaidecagon) – 13 sides
tetradecagon (tetrakaidecagon) – 14 sides
pentadecagon (pentakaidecagon) – 15 sides
hexadecagon (hexakaidecagon) – 16 sides
• heptadecagon (heptakaidecagon)
– 17 sides
• octadecagon (octakaidecagon) – 18 sides
• enneadecagon (enneakaidecagon)
– 19 sides
• Icosagon - 20 sides
•
•
•
•
•
•
•
triacontagon - 30 sides
tetracontagon - 40 sides
pentacontagon - 50 sides
hexacontagon - 60 sides
heptacontagon 70 sides
octacontagon - 80 sides
enneacontagon - 90 sides
•
•
•
•
•
Hectagon – 100 sides
Chiliagon – 1,000 sides
Myriagon – 10,000 sides
Megagon – 1,000,000 sides
Googolgon - 10100 sides
• n-gon – n sides
gon - figure having (so many) angles
Interior of Polygon
Interior Points of Polygon
• Points in the interior of a polygon
Exterior of Polygon
Exterior Points of Polygon
• Points in the exterior of a polygon
Interior Angles Polygons
Interior Angles Polygons
Exterior Angles Polygons
Naming Polygons
• Polygons are named using the
letters at the vertices connecting
adjacent sides of the polygon
(around the polygon)
Diagonals of Polygon
• Line segments joining/connecting
nonadjacent vertices of a polygon
Some Diagonals for Polygons
• Number of diagonals from one
vertex: n - 3
• Number of diagonals:
1
2
n ( n - 3)
• Number of triangles that can be
created from one vertex: n -2
n is the number of sides
vertex: n - 3
Why n – 3 diagonals from one vertex???
vertex: n - 3
Why n – 3 diagonals from one vertex???
n is the number of vertices
vertex: n - 3
vertex: n - 3
No diagonal to
itself
vertex: n - 3
No diagonal to
itself
Subtract 1
from count
vertex: n - 3
No diagonal to
endpoint of
this side
vertex: n - 3
No diagonal to
endpoint of
this side
Subtract 1
from count
vertex: n - 3
No diagonal to
endpoint of
other side
vertex: n - 3
No diagonal to
endpoint of
other side
Subtract 1
from count
vertex: n - 3
There are n vertices but 3
vertices cannot be connected
to another vertex to form a
diagonal within a polygon.
vertex: n - 3
There are n - 3 vertices
which can be connected to
another vertex to form a
diagonal within a polygon.
vertex: n - 3
1
2
n ( n - 3)
vertex: n - 3
Why
1
1
2
n ( n - 3)
n ( n - 3 ) for the total number
2
of diagonals???
vertex: n - 3
Why
1
2
n ( n - 3)
1
n ( n - 3 ) for the total number
2
of diagonals???
1
2
n ( n - 3)
1
2
n ( n - 3)
1
2
n ( n - 3)
When counting
diagonals, this
diagonal …
1
2
n ( n - 3)
When counting
diagonals, this
diagonal is
counted as a
diagonal for
this vertex …
1
2
n ( n - 3)
… as well as
this vertex.
1
2
n ( n - 3)
So, this
diagonal is
counted twice.
1
2
n ( n - 3)
Each vertex has n - 3 diagonals.
1
2
n ( n - 3)
There are n vertices, each of
which has n - 3 diagonals.
1
2
n ( n - 3)
Since the product of n and n - 3
would count each diagonal twice,
we multiply the product, n ( n - 3 ),
by 12 so that each diagonal is
counted only once.
vertex: n - 3
1
2
n ( n - 3)
vertex: n - 3
1
2
n ( n - 3)
Why n – 2 triangles???
Three vertices
are connected
to form a
triangle.
This vertex
alone cannot
form a
triangle.
This vertex
alone cannot
form a
triangle.
Subtract 1
from count
This vertex
alone cannot
form a
triangle.
This vertex
alone cannot
form a
triangle.
Subtract 1
from count
There are n – 2 vertices
form triangles from one
vertex within a polygon.
There are n – 2 vertices
form triangles from one
vertex within a polygon.*
*Note: We are counting non- overlapping adjacent triangles.
vertex: n - 3
1
2
n ( n - 3)
created from one vertex: n -2 *
*Note: We are counting non- overlapping adjacent triangles.
Convex Polygons
Convex Polygon
• A polygon is called convex if
 All diagonals are in the interior
of the polygon
Convex Polygon
of the polygon
Convex Polygon
of the polygon,
 All line segments connecting two
interior points of the polygon are
in the interior of the polygon, and
Convex Polygon
of the polygon,
Convex Polygon
of the polygon,
 All interior angles of the polygon
have measure less than 180º.
Convex Polygon
of the polygon,
 All interior angles of the polygon
have measure less than 180º.
Concave Polygons
Concave Polygon
• A polygon is called concave if
 At least one diagonal contains exterior
points of the polygon
Concave Polygon
points of the polygon
Concave Polygon
points of the polygon,
 At least one line segment connecting
two interior points of the polygon
contains exterior points of the polygon,
and
Concave Polygon
and
Concave Polygon
and
 At least one interior angle of the
polygon has measure greater than
180º.
Concave Polygon
and
 At least one interior angle of the
polygon has measure greater than
180º.
Regular Polygon
• A polygon is called regular if
 all the sides of the polygon have
the same length
and
 all the interior angles of the
polygon have the same measure.
Regular Polygon
the same length
and
This appears
to be a
regular polygon.
Regular Polygon
the same length
and
This is not a
regular polygon.
Regular Polygon
• A regular polygon is
 Equiangular
and
 Equilateral.
Irregular Polygon
• A polygon is called irregular if
 at least one side is longer/shorter
than the other sides of the
polygon
and
 at least one interior angle of the
polygon has a different measure
than the other interior angles.
This does not seem to be
an irregular polygon.
Irregular Polygon
polygon
and
This is an irregular polygon.
Irregular Polygon
polygon
and
Angles of Regular Polygon
• Measures of interior angles:
180(n -2)
degrees
n
• Measures of exterior angles:
360
degrees
n
• Measures of interior angles:

360 
 180  degrees
n 

• Measures of exterior angles:
360
degrees
n
•
•

360 
Measures of interior angles:  180  degrees
n 

360
Measures of exterior angles:
degrees
n
Since the interior angle and exterior angle
at any vertex of a polygon form a straight
angle, the sum of the measure of the
interior angle and the measure of the
exterior angle at any vertex is 180 degrees.
• Measures of interior angles: 180(n -2) degrees
•
n
360
degrees
n
If we would like to determine the sum of the
measures of the interior angles of a regular polygon
or the sum of the measures of the exterior angles of
a regular polygon, we can multiply each formula above
by n to determine the respective sum.
• Measures of interior angles: 180(n -2) degrees
•
n
360
degrees
n
If we would like to determine the sum of the
measures of the interior angles of a regular polygon
or the sum of the measures of the exterior angles of
a regular polygon, we can multiply each formula above
by n to determine the respective sum. That is, …
• Sum of measures of interior angles:
180(n -2) degrees
• Sum of measures of exterior angles:
360 degrees
180(n -2) degrees
Notice the factor of n – 2 above???
180(n -2) degrees
Notice the factor of n – 2 above???
n – 2 is the number of non- overlapping adjacent
triangles that can be formed using diagonals from
one vertex of a polygon.
180(n -2) degrees
Since n – 2 non- overlapping adjacent triangles can be
formed from one vertex of a polygon …
180(n -2) degrees
formed from one vertex of a polygon and since the
sum of the measures of the interior angles of a
triangle is 180 degrees, …
180(n -2) degrees
formed from one vertex of a polygon and since the
sum of the measures of the interior angles of a
triangle is 180 degrees, the sum of the measures of
the interior angles of a polygon formed by these
triangles is 180(n – 2) degrees.

Polygons

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