Polygons
The word
‘polygon’ is a
Greek word.
Poly means
many and
gon means
angles.
Polygons
• The word polygon
means “many angles”
• A two dimensional
object
• A closed figure
Polygons
More about Polygons
• Made up of three or more
straight line segments
• There are exactly two sides
that meet at a vertex
• The sides do not cross each
other
Polygons
Examples of Polygons
Polygons
These are not Polygons
Polygons
Terminology
Side: One of the line
segments that make up a
polygon.
Vertex: Point where two
sides meet.
Polygons
Vertex
Side
Polygons
• Interior angle: An angle
formed by two adjacent
sides inside the polygon.
• Exterior angle: An angle
formed by two adjacent
sides outside the polygon.
Polygons
Exterior angle
Interior angle
Polygons
Let us recapitulate
Exterior angle
Vertex
Side
Diagonal
Interior angle
Polygons
Types of Polygons
• Equiangular Polygon: a
polygon in which all of the
angles are equal
• Equilateral Polygon: a
polygon in which all of the
sides are the same length
Polygons
• Regular Polygon: a
polygon where all the
angles are equal and all
of the sides are the same
length. They are both
equilateral and
equiangular
Polygons
Examples of Regular Polygons
Polygons
A convex polygon: A polygon whose
each of the interior angle measures less
than 180°.
If one or more than one angle in a
polygon measures more than 180° then
it is known as concave polygon. (Think:
concave has a "cave" in it)
Polygons
INTERIOR ANGLES
OF A POLYGON
Polygons
Let us find the connection
between the number of sides,
number of diagonals and the
number of triangles of a
polygon.
Polygons
180
180o
180o
180
o
o
180o
4 sides
Quadrilateral
5 sides
2 x 180o = 360o
2
Pentagon
o
1 diagonal
o
3 x 180 = 540
3
2 diagonals
180o
180o
180o
180o
180o
180o
180o
180o
180o
6 sides
4
Hexagon
4 x 180o = 720o
3 diagonals
7 sides
5
Polygons
Heptagon/Septagon
5 x 180o = 900o
4 diagonals
Regular
Polygon
Triangle
No. of
sides
No. of
diagonals
No. of
3
0
1
Polygons
Sum of the
interior
angles
0
180
Each
interior
angle
0
180 /3
0
= 60
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
0
0
360 /4
0
= 90
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
0
360 /4
0
= 90
Pentagon
5
2
3
3 x180
0
= 540
0
540 /5
0
= 108
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
0
0
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
0
360 /4
0
= 90
Pentagon
5
2
3
3 x180
0
= 540
0
540 /5
0
= 108
Hexagon
6
3
4
4 x180
0
= 720
0
720 /6
0
= 120
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
0
0
0
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
0
360 /4
0
= 90
Pentagon
5
2
3
3 x180
0
= 540
0
540 /5
0
= 108
Hexagon
6
3
4
4 x180
0
= 720
0
720 /6
0
= 120
Heptagon
7
4
5
5 x180
0
= 900
0
900 /7
0
= 128.3
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
0
0
0
0
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
0
360 /4
0
= 90
Pentagon
5
2
3
3 x180
0
= 540
0
540 /5
0
= 108
Hexagon
6
3
4
4 x180
0
= 720
0
720 /6
0
= 120
Heptagon
7
4
5
5 x180
0
= 900
0
900 /7
0
= 128.3
“n” sided
polygon
n
Association
with no. of
sides
Association
with no. of
sides
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
Association
with no. of
triangles
0
0
0
0
Association
with sum of
interior
angles
Regular
Polygon
No. of
sides
No. of
diagonals
No. of
Triangle
3
0
1
180
Quadrilateral
4
1
2
2 x180
0
= 360
0
360 /4
0
= 90
Pentagon
5
2
3
3 x180
0
= 540
0
540 /5
0
= 108
Hexagon
6
3
4
4 x180
0
= 720
0
720 /6
0
= 120
Heptagon
7
4
5
5 x180
0
= 900
0
900 /7
0
= 128.3
“n” sided
polygon
n
n-3
n-2
(n - 2)
0
x180
Polygons
Sum of the
interior
angles
0
Each
interior
angle
0
180 /3
0
= 60
0
0
0
0
(n - 2)
0
x180 / n
1
Calculate the Sum of Interior
Angles and each interior angle of
each of these regular polygons.
7 sides
Septagon/Heptagon
Sum of Int. Angles
900o
Interior Angle 128.6o
2
3
4
9 sides
10 sides
11 sides
Nonagon
Decagon
Hendecagon
Sum 1260o
I.A. 140o
Sum 1440o
I.A. 144o
Sum 1620o
I.A. 147.3o
Polygons
Find the unknown angles below.
w
75o
75o
2 x 180o = 360o
360 – 245 = 115o
140o
x
100o
70o
125o
100o
115o
3 x 180o = 540o
540 – 395 =
145o
125o
z
138o
138o
133o
y
4 x 180o = 720o
720 – 603 =
117o
95o
110o
121o
117o
Diagrams not
drawn
accurately.
105o
137o
5 x 180o = 900o
Polygons
900 – 776 =
124o
EXTERIOR ANGLES
OF A POLYGON
Polygons
An exterior angle of a regular polygon is
formed by extending one side of the polygo
Angle CDY is an exterior angle to angle CDE
B
A
C
F
2
E
1
D
Y
Exterior Angle + Interior Angle of a regular polygon =180
Polygons
0
1200
600
600
600
1200
Polygons
1200
1200
1200
1200
Polygons
1200
1200
1200
Polygons
3600
Polygons
600
600
600
600
600
600
Polygons
600
600
600
600
600
600
Polygons
3
4
600
600
2
600
600
5
600
600
1
6
Polygons
3
4
600
600
2
600
600
5
600
600
1
6
Polygons
3
4
2
3600
5
1
6
Polygons
900
900
900
900
Polygons
900
900
900
900
Polygons
900
900
900
900
Polygons
2
3
3600
1
4
Polygons
No matter what type of
polygon we have, the sum
of the exterior angles is
ALWAYS equal to 360º.
Sum of exterior angles =
360º
Polygons
In a regular polygon with ‘n’ sides
Sum of interior angles = (n -2) x 180
i.e. 2(n – 2) x right
angles
Exterior Angle + Interior Angle =180
0
Each exterior angle = 360 /n
0
No. of sides = 360 /exterior angle
Polygons
0
0
Let us explore few more problems
• Find the measure of each interior angle of a polygon
with 9 sides.
0
• Ans : 140
• Find the measure of each exterior angle of a regular
decagon.
0
• Ans : 36
• How many sides are there in a regular polygon if
each interior angle measures 165 ?
• Ans : 24 sides
• Is it possible to have a regular polygon with an
exterior angle equal to 40 ?
• Ans : Yes
0
0
Polygons
Polygons
DG
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