[worksheet] arc & area of sectors

Length of Arc & Area of sector 1

LENGTH OF ARC AND AREA OF SECTOR

Arc length and the area of a sector are proportional to the angle subtended at the centre of the

circle.

Skill assessed

Calculate the arc length when the radius r and angle are given. Calculate the perimeter of an enclosed shape, involving sectors and triangles. Calculate the area of a shaded region, involving sectors, triangles and segments

Usual format of questions A diagram and other information are given.

By using7

22 = , calculate (a) the perimeter of the whole diagram

(b) the area of the shaded region.

Strategies for problem solving

1. Identify the perimeter of the whole diagram by tracing the outline of the perimeter using

colour.

2. Identify the values of the corresponding radius and angle for the arc length which you want to

find.

3. Calculate the arc length by using the correct formulae.

4. Repeat the process if there is another arc length with different radius and angle.

5. Find the perimeter by adding the arc lengths and the length of straight lines.

6. Identify the region for which you want to find its area.

7. Identify the values of the corresponding radius and angle for the sector.

8. Calculate the area of the shaded region by using the correct formulae.

Common Errors

Students used the wrong formula. Students used wrong values for radius or angle.

Students used other values of like 3.14. Students using the Additional Mathematics method round off the value of angle in radian to 2

or 3 significant figures only.

Students calculated the perimeter of the shaded region when the question asked the perimeter of the whole diagram.

Students did not read the question carefully to extract the right information.

rO

A

B

Arc length of AB = 2360

r

Area of sector AOB = 2r

360


3.1. CIRCUMFERENCE

Example: Use = 7

22

Radius = 7 cm

Circumference

= 2 ( 7 )

= 77

222

= 44 cm

Ex

Example: Diameter = 7 cm

Radius = 3 2

1cm

Circumference = 2 (3 2

1)

= 2 x 7

22 x

2

7

= 22 cm

Exercise A

1.

Radius = 14 cm

Circumference =

Exercise B

1. Diameter = 14 cm

Radius =

Circumference =

2. Radius = 16 cm

Circumference =

2.

Diameter = 140 mm

Radius =

Circumference =

Circumference = 2r, where r is a radius

r

O

7O

O

O14O

O O


3.

Radius = 10 cm

Circumference =

3. Diameter = 35 cm

Radius =

Circumference =

3.2. ARC OF A CIRCLE

Example: Use = 7

22

Radius = 7 cm

Minor arc of AB

= 360

90 2 ( 7 )

= 360

90 2 x

7

22 x 7

= 11 cm

Example: Use = 3.142

Radius = 3 cm

Minor arc of AB =

= 360

602 (3)

= 360

60 2 3.142 3

= 3.142 cm

Exercise A

1.

Radius = 14 cm

= 45o

Minor arc of AB =

Exercise B

1. Radius = 5 cm

= 80o

Minor arc of AB =

OB

A

90o

In the diagram, arc AB subtended an angle at the centre O with radius r.

Length of arc = o360

r 2

O10O

BA

907

O

BA

603

O

BA

45

14

O

BA

80

5

O


7

B

A

1280

O

B

A

r

O

A

B

2. Radius = 21 cm

= 135o Minor arc of AB =

2.

Radius = 20 cm

= 120o Minor arc of AB =

3.

Diameter = 7 cm

Radius =

Major arc of AB =

3. Diameter = 18 cm

= 60o Radius =

Major arc of AB =

3.3. PERIMETER OF SHADED REGION

Example:

Minor arc of AB

= 360

90 2 x

7

22 x 7

= 11 cm

Perimeter of shaded region

= 7 + 7 + 11

= 25 cm

Example:

Minor arc of AB

= 360

80 2 3.142 12

= 16.76


= 16.76 + 12 + 12

= 40.76 cm

60o

AB subtended an angle at the centre O with radius r.

Perimeter of shaded region = Arc AB + OB + OA

= Arc AB + 2r

BA

135

21

O

BA

120

O

BA

OB

A

O


21120

O

B

A

9100

OB

A

7

120C

A

B

10

30C

A

B

7O

7

O

Exercise A

1.

Minor arc of AB

=


=

Exercise B

1.

Minor arc of AB

=


=

2. Major arc of AB =


=

2. Major arc of AB =


=

3.4. AREA OF A CIRCLE

Example: Use = 7

22

Radius = 7 cm

Area = ( 7 )2

= 7

227 7

= 154 cm2

Example:

Diameter = 7 cm

Radius = 32

1 cm

Area = (32

1)2

=

22 7 7

7 2 2

= 1

382 cm

2

Area of a circle = r2

rO


14

O

21

O

31

2O

O

10O O

r

O

A

B

Exercise A

1. Radius = 14 cm

Area =

Exercise B

1. Diameter = 21 cm

Radius =

Area =

Radius = 32

1 cm 2.

Area =

2. Diameter = 140 mm

Radius =

Area=

3.

Radius = 10 cm

Area =

3. Diameter = 35 cm

Radius =

Area =

3.5. AREA OF SECTOR

Area of minor sector AOB

= o360

2 r


7

120C

A

B

Example: Using = 7

22

Radius = 7 cm


= 360

90 x 72

= 360

90

7

22 x 7 x 7

= 38.5 cm 2

Example: Using = 3.142

Radius = 12 cm

Area of minor sector

= 360

80 3.142 x 12 x 12

= 100.544cm2

Exercise 1.

Radius =


=

Exercise B

1.

Radius =


=

2.

Radius =

Area of major sector AOB

=

2.

Radius =

Area of major sector AOB

=

7

B

A

21120

O

B

A

9100

OB

A

10

30C

A

B

1280

O

B

A


3.6 AREA OF SHADED REGION

Example :

Exercise :

1.

2.

3.

7 7 7 7

BAO

14 Area of shaded region =

- 2

= 22 77

22214

7

22

= 308

7

3 3

2 2

2

2

Area of shaded region =

=

10

O

40o


=


-

=

6 3


Questions based on the examination format (Paper 2)

1. In the diagram below, O is the centre of the arc of the circle MNPQ and RSM is a quadrant

with centre P. MOP is a straight line.

Using 7

22= , calculate

a) the perimeter of the whole diagram,

b) the area of the shaded region. [ 6 marks]

Answer :

a) Perimeter of the whole diagram

= 14 77

222

360

180 14

7

222

360

90++

= 58 cm

b) Area of the shaded region = 2 290 22 60 22

14 - 7 360 7 360 7

= 3

1128 cm

2

Exercise 1. In Diagram 1, JKL is arc of circle with centre M. NML is a straight line and JN = NM =

7 cm.

Using 7

22= , calculate

a. the area of the shaded region, in cm2,

b. perimeter of the whole diagram, in cm. [6 marks]

N M L

J

K

Diagram 1

240o

R P

S

M

N

Q

O

14 cm


2. In Diagram 2, O is the centre of the arc of the circle PQR and a quadrant STU. OSR is

a straight line.

Using 7

22= , calculate

a. perimeter of the whole shaded region,

b. area of the whole shaded region. [6 marks]

3. In Diagram 3, OAB, OCD and OEF are three sectors with same centre O.

Given AOF, OCB and ODE are straight lines. Using7

22= , calculate

a. the area of sector OCD,

b. the perimeter of the whole diagram.

[6 marks ]

U

45o

R

P

Q

S

T

O

14 cm

7 cm

Diagram 2

Diagram 3

21 cm

4040

7 cm

OF

E

DC

A

B


4. Diagram 4 shows three quadrants OPQ, TQR and URS. POUS is a straight line and

TOUR is a square.

Using 7

22= , calculate

a) the perimeter of the whole diagram,

b) the area of the whole diagram.

[6 marks ]

5. In Diagram 5, QR and TU are two arc of circles with the same centre O. QPOU and

RSTO are straight lines.

Using 7

22= , calculate

a) ROQ ,

b) area of the shaded sector OTU,

c) perimeter of the whole diagram.

[6 marks]

Diagram 5

Q P O

S

R

T

U 7 cm

Diagram 4

P

Q

R

S

T

U O 14 cm


6. Diagram 6 shows one circle and two semicircles with diameter PQ, QR and PR

respectively. PQR is straight line.

Given that PQ = 3

1 PR and PR = 21 cm. Using

7

22 = , calculate

a) the perimeter in cm, of the shaded region,

b) the area, in cm2, of the whole diagram [6 marks]

7. In diagram 7, O is the centre of the circle with diameter POR = 16 cm. N is midpoint of

radius OR and PMN is a semicircle.

Using 7

22= , calculate

(a) the perimeter ,in cm, of the shaded region.

(b) the area ,in cm2 ,of the shaded region. [6 marks]

P Q R

P

N R

M

O

T S

60o

Diagram 6

Diagram 7


8. In diagram 8, TSR is a quadrant with centre O, P are the centre of the arc of the circle

OVU and a semicircle OQR..

It is given that OT = 20 cm.

Using 7

22= , calculate

a) the perimeter of the whole

diagram.

b) the area of the shaded region.

[ 6 marks]

9. Diagram 9 shows a sector LMN with centre

O and a semicircle OKN.

It is given that OL = 21 cm.

Using 7

22= , calculate

a) the perimeter of the whole diagram.

b) the area of the shaded region.

[ 6 marks]

10. In diagram 10, O is a centre of circle

with diameter KON = 14 cm. KO and

ON are diameter of two semicircles.

Given that MON = 30o.

Using7

22= , calculate

a) the arc of KLM

b) the area of minor sector MON

c) the area of the shaded region.

[6 marks]

P

Q

R

V

U

O T

S

110o

R

Diagram 8

Diagram 9

L

N

O M

K

60o

L

N O M K

M

Diagram 10


Past Year SPM Questions (Paper 2)

1. November 2003

Diagram 1 shows two sectors OMN and OPQ with the same centre O and a quadrant QTO with

centre Q.

OM = 14 cm and QT = 7 cm. Using 7

22= , calculate

d) the perimeter of the whole diagram.

e) the area of the shaded region. [ 6 marks]

2. July 2004

In diagram 2, LK is an arc of a circle with centre P and PQRS is an arc of a circle with centre

O. PORL is a straight line.

PK = 21 cm and OP = 7 cm. Using 7

22 = , calculate

a) the area , in cm2 of the shaded region

b) the perimeter in cm, of the whole diagram. [7 marks ]

P O R L

S

K

Q

60

Diagram 1

Diagram 2

60o

O

P

M N

Q

T


3. November 2004

In diagram 3, PQ and RS are arcs of two different circles with O.

RQ = ST = 7 cm and PO = 14 cm.

Using 7

22= , calculate

(a) the area, in cm2, of the shaded region,

(b) the perimeter , in cm, of the whole diagram. [6 marks]

4. July 2005

Diagram 4 shows two sectors, PQR and TUV, with the same centre O. The angle of each

sector is 270o. OSR is a semicircle with centre V. PTO is a straight line and OP = 14 cm.

Using 7

22 = , calculate

a) the perimeter, in cm, of the whole diagram,

b) the area, in cm2, of the shaded region. [6 marks]

P

R

L S

Q

O

V

U

T

Diagram 3

Diagram 4


5. November 2005

Diagram 5 shows two sectors ORST and OUV with the same centre O. RWO is a semicircle

with diameter RO and RO = 2OV. ROV and OUT are straight lines.

OV = 7 cm and =UOV 60o.

Using 7

22= , calculate

(a) the perimeter , in cm, of the whole diagram,

(b) the area, in cm2 , of the shaded region. [6 marks]

6. July 2006

In diagram 6, QRS and UT are arcs of two circles, centre P and S respectively.

It is given that PUS is a straight line, PQ = 21 cm and US = 14 cm. Using 7

22 = , calculate

a) the area, in cm2, of the shaded region

b) the perimeter in cm, of the shaded region. [6 marks]

P

R

120o

S

Q

45o

U

T Diagram 6

Diagram 5 R

W U

S

W

T

O V


7. November 2006

In Diagram 3, OMRN is a quadrant of a circle with centre O and PQ is an arc of another circle with

centre O.

OMP and ORQ are straight lines.

60POQ = . OM = MP = 7 cm and

Using 22

,7

= calculate

(a) the perimeter, in cm, of the whole diagram

(b) the area, in cm2, of the shaded region.

[6 marks]

8. June 2007

In Diagram 5, ORS is a sector of a circle and PQTU is a semicircle , with centre O respectively. OTS

and OQR are straight lines. The length of arcs PQ, QT and TU are equal .

OS = 2OU and OU = 7 cm .

[ Using 22

,7

= ] , calculate

(a) the perimeter, in cm, of the whole diagram

(b) the area, in cm2, of the shaded region.

[6 marks]

Diagram 3

N

O

Q

R

M P

60o

Diagram 5 Q

O

S R

T

P

60o

U


9. November 2007 Q6

Diagram 3 shows quadrant OST and semicircle PQR, both with centre O.

OS = 21 cm and OP = 14 cm.

[Use 22

,7

= ] , calculate

(a) the area, in cm2, of the shaded region.

(b)the perimeter, in cm, of the whole diagram

[6 marks]

10. Jun 2008, Q6

Diagram 6 shows semicircle ABC, centre O, and a sector of a circle AEF, centre A..

AOFG is a straight line. AO = 14 cm and OF = 7 cm.

Using 22

,7

= ] calculate

(a) the perimeter, in cm, of the coloured region,

(b) the area, in cm2, of the coloured region.

[6 marks]

60o

O S

R

Q

T

P

Diagram 3

O

B

F A C 30o

E

B


11. Nov 2008, Q7

In diagram 7, PQ and RS are arc of two different circles which have the same centre O. OPR is a

straight line.

It is given that oo 60 ROS and 36 POQ == .

Using 22

,7

= calculate

(a) the perimeter, in cm, of the sector ORS,

(b) the area, in cm2, of the coloured region.

[6 marks]

R P

S

O

Q

21 cm

14 cm

Diagram 7


ANSWERS

Chapter 3 Arc Length and Area of Sector

Exercise 3.1

Exercise A 1 88 2

7

4100 cm

3

7

662 cm

Exercise B 1 44 2 440 3 110

Exercise 3.2

Exercise A 1 11 cm 2 49.5 cm 3 5.5.cm

Exercise B 1 6.982 cm 2 41.89 cm 3 9.426 cm

Exercise 3.3

Exercise A 1 86 2

3

143 cm

Exercise B 1 33.71 2 77.60

Exercise 3.4

Exercise A 1 616 2

2

138 cm2

3

7

2314 cm2

Exercise B 1 346.5 cm

2 2 15400 mm

2 3 962.5 cm

2

Exercise 3.5

Exercise A

1 462 cm2 2

3

2102 cm2

Exercise B 1 70.695 cm

2 2 288.02 cm

2

Exercise 3.6 1

7

684 cm2

2

7

125 cm2

3

63

50200 cm2

Questions According to Examination Format

1a 115.5 cm

2 1b 47.23 cm

2a 50 cm 2b 115.5 cm2

3a 171.11 cm2 3b 109.78 cm

4a 72 cm 4b 280 cm2

5a 450

5b 57.75 cm2 5c 55.5 cm

6a 66 cm 6b 269.5 cm2

7a

21

1672 cm

7b

21

1111 cm2

8a

7

682 cm

8b

63

16218 cm2


9a 152 cm 9b 981.75 cm2

10a

3

118 cm

10

b 6

512 cm2

10c

3

2102 cm2

SPM Past Years Questions

SPM2003

a

53.67 cm

b

115.5 cm2

SPM2004J

a

192.5 cm2

b

72 cm

SPM2004

a

248.5 cm2

b

89.67 cm

SPM2005J

a 102 cm

b

423.5 cm2

SPM2005

a

3

264 cm

b

154 cm2

SPM 2006J

a

385 cm2

b

97 cm

SPM 2006 a 146

3 cm

b 589

6 cm

2

SPM 2007J

a

3

157 cm

b

3

1128 cm2

SPM 2007

a 243.83 cm2

b 104.33

SPM 2008J

a 83 b

2

1192

SPM 2008

A 64 b

5

2169

[worksheet] arc & area of sectors

Documents