Download Polygons A polygon is a closed figure that is drawn on a plane. It

Part 5: Polygons
A polygon is a closed figure that is drawn on a plane. It must have at least three sides that are
segments. The segments can intersect only at their endpoints, the segments cannot intersect (cross)
and adjacent segments are not collinear.
PQRST is a polygon. WXYZ is not closed, therefore it is not a polygon. ABCDEF is a polygon, but it is
a concave polygon. We will study only convex polygons. There is a test to determine if a polygon is
concave. In figure ABCDE, choose two points in the interior of the figure. If the segment created from
those two points falls anywhere outside the boundaries of the polygon, then the polygon is concave.
A polygon is named by listing its vertices consecutively in a clockwise or counterclockwise direction.
You may choose any vertex of the polygon to start its name.
Thus PQRST is also TSRQP and STPQR and QRSTP and QPTSR and several other names. There
are actually 10 possible names for this polygon.
Many polygons have special names which you will need to recognize. The names are based on the
number of sides (or angles) of the polygon.
Sides
3
4
5
6
7
8
9
10
12
n
Name
triangle
quadrilateral
pentagon
hexagon
septagon
octagon
nonagon
decagon
dodecagon
n-gon
There is a neat exploration for you to do that will help develop your understanding of the next
theorem, the Polygon Angle-Sum Theorem.
Activity: You will need two sheets of paper for this activity. Copy the following table onto one sheet of
paper.
Polygon
triangle
n number
of sides
3
number of
triangles
formed
1
Sum of the
interior angle
measures
180
quadrilateral
X 180 =
pentagon
X 180 =
hexagon
X 180 =
septagon
X 180 =
octagon
X 180 =
Draw a quadrilateral, pentagon, hexagon, septagon and octagon on the other sheet of paper. (If you
want to draw them a little larger you may put your drawings on two sheets of paper.)
Choose one vertex on each polygon and draw the diagonals from that vertex to all of the other nonadjacent vertices. Count the number of triangles formed within each polygon and record it on the
paper.
You know that each triangle contains 180o. Calculate in the last column of the table the sum of the
interior angle measures by multiplying the number of triangles created by 180. When you are done,
you may go to the next page for a summary of the results.
Your table should look something like the one below:
number of
triangles
formed
1
Sum of the
interior angle
measures
180
Polygon
triangle
n number
of sides
3
quadrilateral
4
2
2 X 180 = 360
pentagon
5
3
3 X 180 = 540
hexagon
6
4
4 X 180 =720
septagon
7
5
5 X 180 =900
octagon
8
6
6x 180 =1080
For each polygon with n sides, there are n-2 triangles formed within the polygon from a single vertex.
Each triangle contains 180o, therefore each polygon contains (n-2)x180o total interior degrees. We
sum this up with a theorem.
Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon is (n-2)180.
Next, watch the video of an example problem that illustrates the Polygon Angle-Sum Theorem.
Example: Solve for x.
This polygon has 6 sides, so its interior angle measures sum to (6-2)(180) = 720o.
x + 5x + 3x + 17 + 4x + 36 + 2x + 5x + 7 = 720
20 x + 60 = 720
20x = 660
X = 33
--------------------------------------------------------------------------------------------------------------------Suppose that a polygon is a regular polygon. A regular polygon is a polygon that is equiangular – all
of its angles have the same measure. A regular polygon also has all of its sides congruent. We use
the polygon angle-sum theorem to find the measure of an interior angle of a regular polygon.
For example, an equiangular triangle has 180o divided amongst its three angles, so each angle
measures
= 60o. A square has 360o divided amongst its four angles, so each angle of a square
measures
= 90o.
How big is the interior angle of a regular 14-gon?
(14 – 2)(180) = 2160o
= 154.2o
How big is the interior angle of a regular 22-gon?
(22-2)(180) = 3600o
o
Let’s examine the exterior angles of a triangle.
The exterior angle of a triangle always forms a linear pair with its corresponding interior angle. By the
Triangle Angle-Sum Theorem we know a + b + c = 180o. The sum of the three exterior angles is
(180-a) + (180 – b)+ (180 – c) = exterior angles sum
540 – a – b – c
540 – (a + b + c)
540 – 180 = 360o
The conclusion is the sum of the exterior angles of the triangle sum to 360 o. This result applies to all
convex polygons in the following theorem.
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360 o.
For the hexagon,
The Polygon Exterior Angle-Sum Theorem is very helpful when trying to find the measure of the
exterior angle of a regular polygon. Remember, regular polygons have all interior angles congruent.
From this information, we can recognize that all exterior angles are also congruent. Often this
knowledge can give us an alternate way to determine the measure of an interior angle.
Example: Find the measures of the exterior and the interior angles of an octagon.
Solution: Sum of the measures of the exterior angles = 360o = 45o
8 sides in an octagon
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So each exterior angle measures 45o in a regular hexagon. Each exterior angle is a linear pair with
the interior angle:
45 + interior angle measure = 180
Interior angle measure = 135o
This answer can be verified by the Polygon Angle-Sum Theorem:
(8 – 2)(180) = 1080 = total interior angle measures
1080 = 135o = measure of an interior angle of a regular octagon
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