This document provides information about polygons, including definitions, terminology, properties of different types of polygons, and formulas relating the number of sides, angles, and diagonals. It defines a polygon as a closed, two-dimensional figure formed by three or more line segments. Regular polygons are introduced as those with all sides the same length and all interior angles the same measure. Formulas are given relating the number of sides of a regular polygon to the sum of its interior angles, the measure of each interior angle, and the number of diagonals and triangles it contains. Interior and exterior angles are defined and their relationships explored. Examples and problems are worked through, such as finding missing angle measures.

1. Polygons The word
‘polygon’ is a
Greek word.
PolyPoly means
many and
gongon means
angles.

2. Polygons
Polygons
• The word polygon
means “many angles”
• A two dimensional
object
• A closed figure

3. 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

4. Examples of Polygons
Polygons

5. These are not Polygons
Polygons

6. Terminology
Side: One of the line
segments that make up a
polygon.
Vertex: Point where two
sides meet.
Polygons

7. Vertex
Side
Polygons

8. • 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

9. Interior angle
Exterior angle
Polygons

10. Let us recapitulate
Interior angle
Diagonal
Vertex
Side
Exterior angle
Polygons

11. 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

12. • 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

13. Examples of Regular Polygons
Polygons

14. 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

15. INTERIOR ANGLES
OF A POLYGON
Polygons

16. Let us find the connection
between the number of sides,
number of diagonals and the
number of triangles of a
polygon.
Polygons

17. Quadrilateral
Pentagon
180
o 180
o
180o 180o
180o
2 x 180o
= 360o
3
4 sides
5 sides
3 x 180
o
= 540
o
Hexagon6 sides
180o 180o
180o
180o
4 x 180o
= 720o
4
Heptagon/Septagon7 sides
180o
180o
180o
180o
180o
5 x 180o
= 900o
5
2
1 diagonal
2 diagonals
3 diagonals
4 diagonalsPolygons

18. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Polygons

19. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Polygons

20. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Pentagon 5 2 3 3 x180
0
= 540
0
540
0
/5
= 108
0
Polygons

21. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Pentagon 5 2 3 3 x180
0
= 540
0
540
0
/5
= 108
0
Hexagon 6 3 4 4 x180
0
= 720
0
720
0
/6
= 120
0
Polygons

22. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Pentagon 5 2 3 3 x180
0
= 540
0
540
0
/5
= 108
0
Hexagon 6 3 4 4 x180
0
= 720
0
720
0
/6
= 120
0
Heptagon 7 4 5 5 x180
0
= 900
0
900
0
/7
= 128.3
0
Polygons

23. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Pentagon 5 2 3 3 x180
0
= 540
0
540
0
/5
= 108
0
Hexagon 6 3 4 4 x180
0
= 720
0
720
0
/6
= 120
0
Heptagon 7 4 5 5 x180
0
= 900
0
900
0
/7
= 128.3
0
“n” sided
polygon
n Association
with no. of
sides
Associatio
n with no.
of sides
Association
with no. of
triangles
Association
with sum of
interiorPolygons

24. Regular
Polygon
No. of
sides
No. of
diagonals
No. of Sum of
the
interior
angles
Each
interior
angle
Triangle 3 0 1 180
0
180
0
/3
= 60
0
Quadrilateral 4 1 2 2 x180
0
= 360
0
360
0
/4
= 90
0
Pentagon 5 2 3 3 x180
0
= 540
0
540
0
/5
= 108
0
Hexagon 6 3 4 4 x180
0
= 720
0
720
0
/6
= 120
0
Heptagon 7 4 5 5 x180
0
= 900
0
900
0
/7
= 128.3
0
“n” sided
polygon
n n - 3 n - 2 (n - 2)
x180
0
(n - 2)
x180
0
/ nPolygons

25. Septagon/Heptagon
Decagon Hendecagon
7 sides
10 sides 11 sides9 sides
Nonagon
Sum of Int. Angles
900
o
Interior Angle 128.6
o
Sum 1260
o
I.A. 140
o
Sum 1440o
I.A. 144
o
Sum 1620o
I.A. 147.3
o
Calculate the Sum of Interior
Angles and each interior angle of
each of these regular polygons.
1
2 43
Polygons

26. 2 x 180o
= 360o
360 – 245 = 115o
3 x 180o
= 540o
540 – 395 =
145o
y117o
121o
100o
125o
140o z
133o
137o
138o
138o
125o
105o
Find the unknown angles below.
Diagrams not
drawn
accurately.
75o
100o
70o
w
x
115o
110o
75o
95o
4 x 180o
= 720o
720 – 603 =
117o
5 x 180o
= 900o
900 – 776 =
124oPolygons

27. EXTERIOR
ANGLES OF A
POLYGON
Polygons

28. An exterior angle of a regular polygon is
formed by extending one side of the polygon.
Angle CDY is an exterior angle to angle CDE
Exterior Angle + Interior Angle of a regular polygon =180
0
DE
Y
B
C
A
F
12
Polygons

29. 1200
1200
1200
600
600
600
Polygons

30. 1200
1200
1200
Polygons

31. 1200
1200
1200
Polygons

32. 3600
Polygons

33. 600
600
600
600
600
600
Polygons

34. 600
600
600
600
600
600
Polygons

35. 1
2
3
4
5
6
600
600
600
600
600
600
Polygons

36. 1
2
3
4
5
6
600
600
600
600
600
600
Polygons

37. 1
2
3
4
5
6
3600
Polygons

38. 900
900
900
900
Polygons

39. 900
900
900
900
Polygons

40. 900
900
900
900
Polygons

41. 1
23
4
3600
Polygons

42. 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

43. In a regular polygon with ‘n’ sides
Sum of interior angles = (n -2) x 180
0
i.e. 2(n – 2) x right
angles
Exterior Angle + Interior Angle =180
0
Each exterior angle = 360
0
/n
No. of sides = 360
0
/exterior angle
Polygons

44. Let us explore few more problems
• Find the measure of each interior angle of a polygon
with 9 sides.
• Ans : 140
0
• Find the measure of each exterior angle of a regular
decagon.
• Ans : 36
0
• How many sides are there in a regular polygon if
each interior angle measures 1650
?
• Ans : 24 sides
• Is it possible to have a regular polygon with an
exterior angle equal to 400
?
• Ans : Yes
Polygons

45. Polygons DG

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