Polygons and Angles - west.cdn.mathletics.comwest.cdn.mathletics.com/IWB/Book/393/58494325.CAN_J_Polygons_… · Name these polygons (based on the number of sides) and state whether

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Polygons and Angles

Polygons and Angles

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Polygons & Angles

SERIES TOPIC

J 9

A shape which has straight sides only (no curved sides) is called a Polygon. The angles inside and outside the polygon can be found based only on the number of sides in the polygon. This chapter shows some tricks to find these angles.

POLYGONS & ANGLES

What do I know now that I didn’t know before?

Try to answer these questions, before working through the chapter.

I used to think:

Answer these questions, after working through the chapter.

But now I think:

Sketch a shape which is a polygon and a shape which isn’t a polygon

If a polygon has 6 sides, then how many interior angles does it have?

How do you find the interior angles of a regular polygon?

Sketch a shape which is a polygon and a shape which isn’t a polygon

If a polygon has 6 sides, then how many interior angles does it have?

How do you find the interior angles of a regular polygon?

2 100% Polygons & Angles


Polygons & Angles

SERIES TOPIC

J 9

BasicsPolygons & Angles

A polygon is a closed shape whose sides are all straight.

A triangle is a polygon with three sides and a quadrilateral is a polygon with four sides. Polygons have different names depending on how many sides they have. The table below shows some polygon names depending on the number of sides.

If a polygon has equal angles and equal sides, then it is called a regular polygon. If any of the sides or angles in a polygon are not equal then the polygon is irregular.

Basics

Number of Sides

Polygon Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Undecagon

12 Dodecagon

A polygon can also be convex or non-convex (concave). A polygon is convex if all its interior angles are less than 180c .

This IS a polygon

The shape is closed and all the sides are straight.

This shape is not closed. Not all the sides are straight.

This IS NOT a polygon This IS NOT a polygon

Regular Triangle Regular Pentagon Irregular Quadrilateral

(3 sides, all equal)(Equilateral Triangle)

Convex Polygon

All interior angles are less than 180c .

One interior angle greater than 180c .

Two interior angles are greater than 180c .

Concave PolygonConcave Polygon

(5 sides, all equal) (4 sides, not all equal)

Different Polygons



Polygons & Angles

SERIES TOPIC

J 9

Questions Basics

1. Circle the shapes that are polygons:

2. Name these polygons (based on the number of sides) and state whether each is regular or irregular:

3. Name these polygons (based on the number of sides) and state whether they are convex or concave:



Polygons & Angles

SERIES TOPIC

J 9

Knowing More

Draw a square inside a circle

Step 1: A square is a regular polygon with 4n = sides.

Step 2: 360 n 360 4 90' '= =c c c

Step 3: Draw 4n = radii from the centre of the circle with 90c between each radius.

Draw a regular hexagon inside a circle

Step 1: A regular hexagon has 6n = sides.

Step 2: 360 n 360 6 60' '= =c c c

Step 3: Draw 6n = radii from the centre of the circle with 60c between each radius.

Drawing regular polygons can be quite difficult, so circles are used to make it easier. This is done in 3 easy steps:

• Step 1: Find n , the number of sides in the polygon.

• Step 2: Calculate 360 n'c .

• Step 3: Draw n radii from the centre of the circle with the angle calculated in Step 2 between each radius.

60c

60c

60c

60c

60c

60c

How do I Draw a Regular Polygon?



Polygons & Angles

SERIES TOPIC

J 9

Questions Knowing More

1. Use the circle below to draw an equilateral triangle (regular polygon with 3 sides).

2. Use the circle below to draw a regular pentagon.

3. Use the circle below to draw a regular octagon.



Polygons & Angles

SERIES TOPIC

J 9

Using Our Knowledge

Find the size of x in the following triangle:

62 55 180

117 180

180 117

x

x

x

x 63

`

`

c

c c

c c

c

+ + =

+ =

= -

=

c c

The sum of the interior angles of a triangle is always 180c .

Find the size of x below:

94 91 75 360

260 360

x

x

x 100

`

`

c c c c

c c

c

+ + + =

+ =

=

Find the sizes of x and y below:

( 10 ) 100 180

2 70

35

x x

x

x

c c c

c

c

+ + + =

=

=

360 290

70

y

y

y

85 100 105 360

`

c c c c

c c

c

+ + + =

= -

=

Any quadrilateral can be divided into two triangles by drawing a diagonal. Each triangle has an angle sum of 180c .

Sum of interior angles of a quarilateral = 2 # Angle sum of a triangle

= 2 # 180c

= 360c

62c

x

55c

Angle sum 180c

Angle sum 180c

91c

75c

94c

x

100c

10x c+ x

y

40c 70c

(Angle sum of a quadrilateral is 360c )

(Angle sum of a triangle is 180c )

(Angle sum of a quadrilateral is 360c )


Interior Angles of a Triangle

Interior Angles of a Quadrilateral



Polygons & Angles

SERIES TOPIC

J 9

Using Our Knowledge

For example, a pentagon (5 sides) is divided into triangles like this

Sum of interior angles of a pentagon Angle sumof a triangle3

3 180

540

c

c

#

#

=

=

=

Any polygon can be divided into triangles by drawing diagonals from a common vertex.

So far we know that:

• A polygon with 3 sides (triangle) can be divided into 1 triangle

• A polygon with 4 sides (quadrilateral) can be divided into 2 triangles

• A polygon with 5 sides (pentagon) can be divided into 3 triangles

So each polygon can be divided into a number of triangles that is 2 less than the number of sides in the polygon. This means that a polygon with n sides can be divided into (n - 2) triangles.

If the polygon is a regular polygon then all its angles are equal. So each interior angle will be equal to:

... Sum of interior angles of a polygon with n sides = (n - 2) # 180c

180

n

n 2 # c-^ h

Find the sum of the interior angles of an octagon

An octagon has 8 sides: ... n = 8

Sum of interior angles = (n - 2) #180c

= (8 - 2) # 180c

= 1 800 c

Find the size of each interior angle of a regular decagon

A decagon has 10n = sides. So each angle is equal to:

n

n 2 180

10

10 2 180

144

# #c c

c

-=

-

=

^ ^h h

180c

180c

180c

common vertex

Interior Angles of a Polygon with More Than Four Sides



Polygons & Angles

SERIES TOPIC

J 9

Using Our KnowledgeQuestions

1. Find the value of each of x in each of the following:

a

c

e

b

d

f

x

60c

95c

39c123c

x118c

62c

110c

x

30c

x

2x

x48c 120c

2x

6x

4x

3x



Polygons & Angles

SERIES TOPIC

J 9

Questions Using Our Knowledge

2. Divide this hexagon into 4 triangles by drawing diagonals from a common vertex:

3. Complete the following table for polygons with different sides.

4. Find the sum of interior angles of a polygon with:

a 17 sides b 27 sides

Number of sides n^ h Number of triangles formed 2n -^ h Sum of interior angles n 2 180# c-^ h

4 4 2 2- = 4 2 180 360# c c- =^ h5 5 2 3- =

6

7

8

9

10

11

12



Polygons & Angles

SERIES TOPIC

J 9

Using Our KnowledgeQuestions

5. Answer these questions about the following polygon:

a How many sides does this polygon have?

b What is the sum of the interior angles of this polygon?

c Find the size of x.

6. Find the number of sides of a polygon (n) which has a sum of interior angles of:

a 2880c b 3600c

7. What is the size of each angle in each of the following polyons?

a regular heptagon b regular undecagon

155c

167c

130c 126c

134c

x

162c 148c

146c

160c



Polygons & Angles

SERIES TOPIC

J 9

Thinking More

... The exterior angle is equal to the sum of the interior opposite angles

The exterior angle of a triangle is equal to the sum of the 2 interior opposite angles.

proof

Find the size of +CBD in the following diagram:

CBD BAC ACB+ + += +

CBD

CBD

35 70

105

`

`

c c

c

+

+

= +

=

Find the size of +FEG in the following diagram

EGH FEG EFG+ + += +

FEG

FEG

122 44

122 44

78

`

`

c c

c c

c

+

+

= +

= -

=

ABC BAC ACB

ACB ABC BAC

180

180

c

c

+ + +

+ + +

+ + =

= - -

( )

ACB ACD

ACD ACB

ABC BAC

ABC BAC

180

180

180 180

`

c

c

c c

+ +

+ +

+ +

+ +

+ =

= -

= - - -

= +


(Angles on a str. line are supplementary)

x

y x y+

Interior opposite

angles

Exterior angle

B

B

C

C

A

A 35c

70c

D

D

F

G

HD

E

44c

122c

(Exterior angle of a triangle is equal to the sum of the 2 interior opposite angles)

(Exterior angle of a triangle is equal to the sum of the 2 interior opposite angles)

Exterior Angles of a Triangle



Polygons & Angles

SERIES TOPIC

J 9

Thinking More

The sum of the exterior angles of any polygon is 360c .

So a + b + c + d + e + f = 360c

Find x below

Find the size of +DFG below

a

b

c

d

e

f

85c

x

147c

C

A

H

F

G

B

77c

93c

120c

E

DFE

DFE

DFE

DFG DFE

DFG

DFG

120 77 93 360

290 360

70

180

70 180

70

`

`

`

`

c c c c

c c

c

c

c c

c

+

+

+

+ +

+

+

+ + + =

+ =

=

+ =

+ =

=

85 147 360

232 360

x

x

128

c c c

c c

c

+ + =

+ =

=

(Sum of exterior angles is 360c )

(Angles on a str. line are supplementary)

(Sum of exterior angles is 360c )

Exterior Angle Sum of A Polygon

This is true for any polygon whether it is convex or concave, regular or irregular.



Polygons & Angles

SERIES TOPIC

J 9

Thinking More

Remember, a polygon has the same number of exterior angles (and interior angles) as the number of sides it has. For example, a pentagon has 5 sides, 5 interior angles and 5 exterior angles.

In other words, a polygon with n sides has n exterior angles (and n interior angles). If a regular polygon has n sides then its n exterior angles must all be equal.

Find the size of the exterior angles of a regular hexagon

A hexagon has 6 sides. ... n = 6.

So the exterior angles of a hexagon are equal to:

6

360 60c c=

A regular polygon has equal interior angles of 40c

a Find the numer of sides.

Use the size of the exterior angles to find n, the number of sides:

9

n

n

n

360 40

40360`

`

c c

cc

=

=

=

A polygon with n = 9 sides is called a nonagon.

b Find the size of the interior angles.

The size of the interior angles of a regular nonagon is given by:

n

n 2 180

9

9 2 180

91260

140

# #c c

c

c

-=

-

=

=

^ ^h h

Exterior Angles of Regular Polygons

A regular polygon with n sides has exterior angles all equal to n

360c



Polygons & Angles

SERIES TOPIC

J 9

Thinking MoreQuestions

1. Use the exterior angles of a triangle to find x in the following diagrams:

a

c

e

b

d

x

70c 50c

x 33c

80c

x

137c

2x

132cx

C

D

B

E

FA

x

2x17c

110c



Polygons & Angles

SERIES TOPIC

J 9

Questions Thinking More

2. Find x in the following diagrams:

a

b

55c 17c

62c

86c

x

139c

66c

x

78c

125c

130c

27c

158c

x

2x



Polygons & Angles

SERIES TOPIC

J 9

Thinking MoreQuestions

3. What size are the exterior angles of a regular quadrilateral (polygon with 4 sides)?

4. What size are the exterior angles of a regular decagon?

5. For a regular polygon with 15 sides:

a What size are the exterior angles?

b What size are the interior angles?

c What is the sum of the interior angles?

6. A regular polygon has exterior angles which are 18c.

a How many sides this polygon have?

b What size are the interior angles?

c What is the sum of the interior angles?



Polygons & Angles

SERIES TOPIC

J 9

Answers

Basics: Basics:

Knowing More:

1.

1.

2.

2.

3.

Regular Octagon

Irregular quadrilateral

Irregular Pentagon

Irregular dodecagon Irregular pentagon.

Regular Quadrilateral (square)

Irregular undecagon.

Irregular triangle

Quadrilateral (convex) Triangle (convex)

Decagon (concave) Heptagon (concave)

Octagon (concave) Triangle (convex)

Quadrilateral (concave) Nonagon (concave)

120c 120c

120c

72c 72c

72c

72c

72c



Polygons & Angles

SERIES TOPIC

J 9

Answers

3.

1.

1.

2.

3.

4.

5.

6.

2.

3.

4.

5.

6.

7.

45c45c

45c

45c45c

45c

45c

45c

Knowing More:

Using Our Knowledge:

Using Our Knowledge:

Thinking More:

25x = c x 80= c

a

a

a

a

a

c

e

a

a

a

a

b

b

b

b

b

b

b

c

c

b

d

b

c

c dx 82= c x 75= c

f x 24= ce x 24= c

Num

ber

of

side

s n^h

Num

ber

of tr

iang

les

form

ed

2n-

^h

Sum

of i

nter

ior

angl

es

n2

180

#c

-^

h

44

22

-=

42

180

360

#c

c-

=^

h

55

23

-=

(52)

180

540

#-

=c

c

66

24

-=

(62)

180

720

#-

=c

c

77

25

-=

(7

2)

180

900

#-

=c

c

88

26

-=

(2)

180

08

108

#-

=c

c

99

27

-=

(2)

180

10

926

#-

=c

c

10

10

28

-=

(2)

180

10

10

44

#-

=c

c

11

11

29

-=

(2)

180

10

11

62

#-

=c

c

12

12

210

-=

(2)

180

10

12

80

#-

=c

c

2700c 4500c

10

1440c

n 18= n 22=

( d.p.)147.3 1c( d.p.)128.6 1c

112x = c

x 120= c x 47= c

x 133= c x 44= c

x 31= c

x 33= c x 32= c

90c

36c

24c 156c

20 162c

3240c

2340c



Polygons & Angles

SERIES TOPIC

J 9

Notes



Polygons & Angles

SERIES TOPIC

J 9

Notes

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Polygons and Angles

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Polygons and Angles - west.cdn.mathletics.comwest.cdn.mathletics.com/IWB/Book/393/58494325.CAN_J_Polygons_… · Name these polygons (based on the number of sides) and state whether