Mohd Khusaini Majid

Mrsm 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 ABCDE

has 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

B

Solution:

ABCD is not a regular polygon

because A ≠ B.

b) BA

H

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°

= 60°

3

b)

100°

2x110°

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°

75°c

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

120°110°

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

85°136°

125°

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° = 117°

2

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° = 1440°

n – 2 = 1440°

= 8

n = 10

(b) Let n be the number of sides

of the polygon

(n – 2) x 180° = 1080°

n – 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) Pentagon

b) Heptagon

c) Decagon

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

interior angles is

a) 720°

b) 900°

c) 1260°

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

a)

130°

x

140°

135°

144°

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

65°60°

ED

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°

60°

80°

75°

150°

x

x

x

c) d)

110°74°

140°

150°100°

68°

75° y

x

z

w

z

r

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 3°7

7

ANOTHER WAY: Exterior angle = 180° – Interior angle

= 180° – 128 4° = 51 3°7 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 = 140n

180n - 140n = 360

40n = 360

Thus, n = 360

= 9

n

Another Way: Interior angle = 140°

Exterior angle = 180° - 140°

= 40°

Hence, 360° = 40°

n = 9

n

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) Pentagon

b) Octagon

c) Hexagon

d) Decagon

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

that its

a) interior angle is 135°

b) interior angle is 108°

c) exterior angle is 36°

d) 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 problem

Given : One square and two hexagons with sides of

the same length

Find : Sum of one interior angle of a square and one

interior angle of each hexagon

Devising a strategy

Find 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 strategy

Interior 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° - 90°

2

(n – 2) x 180° = 135°

180n – 360 = 135n

45n = 360

n = 360 = 8

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

octagon.

Checking the answer

Use 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. Calculate

a) the number of sides

b) the sum of the interior angles

of the regular polygon FBCG.

GF

D

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.