Mohd Khusaini MajidMrsm Kota Kinabalu

2.1 CONCEPT OF REGULAR

POLYGONS

An equilateral triangle has equal sides and equal interior angles. Thus, AB = BC = CA and

A = B = C = 60

A

A

A

C B

D C

B

E

D C

B

A square has equal sides and equal interior angles. Thus, AB = BC = CD = DA and

A = B = C = D = 90

Is all sides of pentagon ABCDEhas the same length and the angles are of the same size?

RECALL

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

The sum of the interior angles of a square is 360.

A regular polygon is a polygon in which

a) all sides are of equal length and

b) all interior angles are of equal size.

Example 1Example 1Example 1Example 1Determine if each of the polygons below is a regular polygon. Give your reason if it is not a regular polygon.

a) A

D C

BSolution:

ABCD is not a regular polygon because A B.

b) BAH

G

F

C

D

E

Solution:

ABCDEFGH is a regular polygon.

Test YourselfTest YourselfTest YourselfTest YourselfDetermine if the following are regular polygon. Give your reason if it is not a regular polygon.

1. L

M N

4.3.

2. SP

RQ

U

T

S

R VW

VU

T

3 cm

3 cm 3 cm

3 cm

Copy the following polygons. Draw all the axes of symmetry of each polygon if there are any. State the number of axes of each polygon.

Exercise 2.1AExercise 2.1AExercise 2.1AExercise 2.1A

1.

4.3.

2.

Find the size of interior and exterior angles

2.2 EXTERIOR AND INTERIOR

ANGLES OF POLYGONS

exterior angle

interior angle

In a polygon, the interior and exterior angles lie on a straight line.

Interior angle + Exterior angle = 180

Example 4Example 4Example 4Example 4Find the values of x and y in the following polygons.a)

x

105

2y y

Solution:

x + 105 = 180

x = 180 105

= 75

2y + y = 180

3y = 180

y = 180

= 603

b)

1002x

110

y

Solution:

2x + 100 = 180

2x = 180 100

= 80

x = 80

= 40

y + 110 = 180

y = 180 - 110

= 70

2

Find the values of the unknown angles in each polygons below.

Exercise 2.2AExercise 2.2AExercise 2.2AExercise 2.2A

1.b

48

a132

2.110

75c

d

f

Determine the sum of the interior angles of a polygon

RECALL

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

The sum of the interior angles of a square is 360.

What is the sum of the interior angles of a pentagon, hexagon and other polygons?

The sum of the interior angles of a polygon with n sides is (n 2) x 180

Example 5Example 5Example 5Example 5Find the value of x in each of the polygons below.a)

95

x

120110

Solution:

The sum of the interior angles of a pentagon = (5 2) x 180= 3 x 180

= 540

x + 90 + 120 + 95 + 110 = 540

x + 415 = 540

x = 540 415 = 125

Use (n 2) x 180

b)140

x

x

85136125

Solution:

The sum of the interior angles of a hexagon = (6 2) x 180= 4 x 180

= 720

x + x + 140 + 125 + 136 + 85 = 720

2x + 486 = 720

2x = 720 486

x = 234 = 1172

Example 6Example 6Example 6Example 6

Find the number of sides of a polygon if the sum of its interior angles is(a) 1440 (b) 1080

(a) Let n be the number of sides of a polygon.

(n 2) x 180 = 1440n 2 = 1440

= 8

n = 10

(b) Let n be the number of sides of the polygon

(n 2) x 180 = 1080n 2 = 1080

= 6

n = 8

Solution:

180 180

Exercise 2.2BExercise 2.2BExercise 2.2BExercise 2.2B

1. Find the sum of the interior angles of each of the following polygons.

a) Pentagonb) Heptagonc) Decagon

2. Find the number of sides of a polygon if the sum of its interior angles is

a) 720b) 900c) 1260

3. Find the value of x in each of the polygons below.a)

130x

140

135144

160

b)60

x

x

4. The diagram below shows a hexagon. Find the value of x + y.

70

y

yx

x

Determine the sum of the exterior angles of a polygon

The sum of the exterior angles of a polygon is 360.

B

A

D

C

Example 7Example 7Example 7Example 7Find the values of the unknown angles in each of the polygons below.a)

40

y z

x 75

Solution:

x = 180 75

= 105

y = 360 (40 + 90 + 105)= 360 235

= 125

z = 180 125

= 55

Supplementary angles

Sum of the exterior angles of a polygon is 360

Supplementary angles

b)

75

y

x3x

6560

E D

C

BA

Solution:

x + 3x = 180

4x = 180

x = 45

Extend the side EA.

Exterior angle of A = 180 75

= 105

y = 360 (60 + 45 + 105 + 65)= 360 275

= 85 Sum of the exterior angles of a polygon is 360

Exercise 2.2CExercise 2.2CExercise 2.2CExercise 2.2C

1. Calculate the unknown angles in the following polygons.

a) b)112

45

6080

75

150

x

x

x

c) d)

11074140

150100

68

75 y

x

z

w

zr

s

r

p

q

Find the interior angles, exterior angles and number of sides of a regular polygon

A regular polygon has equal interior angles, equal exterior angles and sides of equal length.

The sum of the interior angles of a polygon with n sides is (n 2) x 180.

Thus, each interior angle of a regular polygon is (n 2) x 180

n

The sum of the exterior angles of a polygon is 360.

Thus, each exterior angle of a polygon is 360

n

Notes

If exterior angle = 360 , then

interior angle = 180 - 360 .n

n

Example 8Example 8Example 8Example 8

Find the size of the interior angle and the exterior angle of a regular heptagon.Solution:

A regular heptagon has 7 sides.

Sum of the interior angles = (7 2) x 180= 900

Each interior angle = 900

= 128 4

7

7

Each exterior angle = 360

= 51 37

7

ANOTHER WAY: Exterior angle = 180 Interior angle

= 180 128 4 = 51 37 7

Example 9Example 9Example 9Example 9Find the number of sides of a regular polygon given that(a) the exterior angle is 72 (b) the interior angle is 140

Solution:

(a) Let n be the number of sides of the polygon.360 = 72

Thus, n = 360

= 5

n

72

(b) Let n be the number of sides of the polygon.(n 2) x 180 = 140

180n 360 = 140n180n - 140n = 360

40n = 360Thus, n = 360

= 9

n

Another Way: Interior angle = 140

Exterior angle = 180 - 140

= 40

Hence, 360 = 40

n = 9n

40

Exercise 2.2DExercise 2.2DExercise 2.2DExercise 2.2D1. Find the size of the interior and exterior angles of the

following regular polygons.a) Pentagonb) Octagonc) Hexagond) Decagon

2. Find the number of sides of a regular polygon, given that its

a) interior angle is 135b) interior angle is 108c) exterior angle is 36d) exterior angle is 120

Solve problems involving angles and sides of polygons

Example 10Example 10Example 10Example 10

Amin is given a square tile and two regular hexagonal tiles. All the tiles have sides of equal length. Determine if he can form a tessellation with these tiles. If Amin must use the square tile, find two other tiles which can tessellate with the square tile.

Solution:

Understand the problemGiven : One square and two hexagons with sides of

the same lengthFind : Sum of one interior angle of a square and one

interior angle of each hexagon

Devising a strategyFind the interior angles of the three polygons.Add to see if the sum of the three interior angles mentioned above 360.

Stage 1

Stage 2

Carrying out the strategyInterior angle of a square is 90.Interior angle of a hexagon is 180 - 360 = 120

Sum of interior angles of the square and two hexagons is 90 + (2 x 120) = 330. Thus, the three tiles do not tessellate.If Amin has to use the square tile and needs to find two tiles which can tessellate with it, each interior angle of the other two tiles is 360 - 90 = 135.

6

2

Stage 3

The sum of the interior angles of the square tile and the two other tiles must be 360. Thus, 90 + (2 x interior angle) = 360

Interior angle = 360 - 902

(n 2) x 180 = 135

180n 360 = 135n45n = 360

n = 360 = 8

Thus, the other two tiles should be in the shape of an octagon.

Checking the answerUse the strategy of working backwards.If two octagonal tiles are used, each interior angle is 135.Sum of the two interior angles of the two tiles is 2 x 135 = 270.To tessellate, the interior angle of the third polygon is 360 - 270 = 90. Thus, a square tile is needed to tessellate with two octagonal tiles.

n

45

Stage 4

Exercise 2.2EExercise 2.2EExercise 2.2EExercise 2.2E1. In the diagram, ABCD is part of a regular decagon.

FBCG is part of a regular polygon. Calculatea) the number of sidesb) the sum of the interior anglesof the regular polygon FBCG.

GFD

CB

A

4

SUMMARY POLYGONS IIPOLYGONS IIPOLYGONS IIPOLYGONS II

Regular polygon

A polygon in which all the sides are of equal length and all the interior angles are of equal size

Irregular polygon

A polygon in which not all the sides are of equal length or not all the interior angles are of equal size

Equilateral triangle

Square Regular pentagon

Regular hexagon

Scalene triangle Rectangle Parallelogram

Exterior angle and interior angle

Interior angle + Exterior angle = 180

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

The sum of the interior angles of a polygon with n sides is (n 2) x 180.

interior angle

exterior angle

The interior angle of a regular polygon with n sides is (n - 2) x 180 .

The exterior angle of a regular polygon with n sides is 360 .

n

n

Axis of symmetry

The number of axes of symmetry of a regular polygon is equal to its number of sides.