Polygons The word ‘polygon’ is

a Greek word.

PolyPoly means many

and gongon means angles.

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

Interior angle

Exterior angle

Polygons

Let us recapitulate

Interior angle

Diagonal

Vertex

Side

Exterior 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

QuadrilateralPentagon

180o 180

o

180o

180o

180o

2 x 180o = 360o 3

4 sides5 sides

3 x 180o = 540o

Hexagon6 sides

180o

180o

180o

180o

4 x 180o = 720o

4 Heptagon/Septagon7 sides

180o

180o180o

180o

180o

5 x 180o = 900o 5

2

1 diagonal2 diagonals

3 diagonals 4 diagonals Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Pentagon 5 2 3 3 x1800

= 5400

5400/5

= 1080

Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Pentagon 5 2 3 3 x1800

= 5400

5400/5

= 1080

Hexagon 6 3 4 4 x1800

= 7200

7200/6

= 1200

Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Pentagon 5 2 3 3 x1800

= 5400

5400/5

= 1080

Hexagon 6 3 4 4 x1800

= 7200

7200/6

= 1200

Heptagon 7 4 5 5 x1800

= 9000

9000/7

= 128.30

Polygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Pentagon 5 2 3 3 x1800

= 5400

5400/5

= 1080

Hexagon 6 3 4 4 x1800

= 7200

7200/6

= 1200

Heptagon 7 4 5 5 x1800

= 9000

9000/7

= 128.30

“n” sided polygon

n Association with no. of

sides

Association with no. of sides

Association with no. of triangles

Association with sum of interior

anglesPolygons

RegularPolygon

No. of sides

No. of diagonal

s

No. of Sum of the

interior angles

Each interior angle

Triangle 3 0 1 1800

1800/3

= 600

Quadrilateral

4 1 2 2 x1800

= 3600

3600/4

= 900

Pentagon 5 2 3 3 x1800

= 5400

5400/5

= 1080

Hexagon 6 3 4 4 x1800

= 7200

7200/6

= 1200

Heptagon 7 4 5 5 x1800

= 9000

9000/7

= 128.30

“n” sided polygon

n n - 3 n - 2 (n - 2) x180

0(n - 2) x180

0 / n

Polygons

Septagon/Heptagon

Decagon Hendecagon

7 sides

10 sides 11 sides9 sides

Nonagon

Sum of Int. Angles 900o

Interior Angle 128.6o

Sum 1260o

I.A. 140oSum 1440o I.A. 144o

Sum 1620o I.A. 147.3o

Calculate the Sum of Interior Angles and each interior angle of each of these regular polygons.

1

2 43

Polygons

2 x 180o = 360o

360 – 245 = 115o

3 x 180o = 540o540 – 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 = 720o720 – 603 = 117o

5 x 180o = 900o900 – 776 = 124oPolygons

EXTERIOR ANGLES OF A

POLYGON

Polygons

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 =1800

DEY

B

C

A

F

12

Polygons

1200

1200

1200

600 600

600

Polygons

1200

1200

1200

Polygons

1200

1200

1200

Polygons

3600

Polygons

600

600

600

600

600

600

Polygons

600

600

600

600

600

600

Polygons

1

2

3

4

5

6

600

600

600

600

600600

Polygons

1

2

3

4

5

6

600

600

600

600

600 600

Polygons

1

2

34

5

6

3600

Polygons

900

900

900

900

Polygons

900

900

900

900

Polygons

900

900

900

900

Polygons

1

23

4

3600

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 1800

i.e. 2(n – 2) x right angles Exterior Angle + Interior Angle =1800

Each exterior angle = 3600/n

No. of sides = 3600/exterior angle

              

Polygons

Let us explore few more problems• Find the measure of each interior angle of a

polygon with 9 sides.• Ans : 1400

• Find the measure of each exterior angle of a regular decagon.

• Ans : 360

• 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

Polygons DG