Chapter 3 II Arc Length n Sector Area ENHANCE

CHAPTER 3: 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 questionsA diagram and other information are given.

By using , calculate (a) the perimeter of the whole diagram

(b) the area of the shaded region.

Strategies for problem solving1. 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.

Length of Arc & Area of sector 1

rOA

B

Arc length of AB =

Area of sector AOB =

3.1. CIRCUMFERENCE

Example: Use =

Radius = 7 cm Circumference = 2 ( 7 )

=

= 44 cm

ExExample: Diameter = 7 cm

Radius = 3 cm

Circumference = 2 (3 )

= 2 x x

= 22 cm

Exercise A1.

Radius = 14 cmCircumference =

Exercise B1. Diameter = 14 cm

Radius = Circumference =

2. Radius = 16 cmCircumference =

2. Diameter = 140 mm Radius = Circumference =

3. Radius = 10 cm Circumference =

3. Diameter = 35 cmRadius = Circumference =


Circumference = 2r, where r is a radiusr

O

7O

O

O14O

O O

O10O

3.2. ARC OF A CIRCLE

Example: Use =

Radius = 7 cmMinor arc of AB

= 2 ( 7 )

= 2 x x 7

= 11 cm

Example: Use = 3.142

Radius = 3 cm Minor arc of AB =

= 2 (3)

= 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 =

2. Radius = 21 cm = 135o

Minor arc of AB =

2. Radius = 20 cm = 120o

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 =

BA

907

O

BA

603

O

BA

4514

O

BA

805

O

BA

13521

O

BA

120

O

7

B

A

12 80

O

B

A

21120

O

B

A

9100

OB

A

r

O

A

B

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

= 2 x x 7

= 11 cm

Perimeter of shaded region = 7 + 7 + 11 = 25 cm

Example:

Minor arc of AB

= 2 3.142 12

= 16.76

Perimeter of shaded region= 16.76 + 12 + 12= 40.76 cm

Exercise A

1. Minor arc of AB =

Perimeter of shaded region =

Exercise B

1. Minor arc of AB

=



60o

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

Perimeter of shaded region = Arc AB + OB + OA = Arc AB + 2r

BA

OB

A

O

7

120C

A

B

10

30C

A

B

7O

7

O

2. Major arc of AB =


2. Major arc of AB =


3.4. AREA OF A CIRCLE

Example: Use =

Radius = 7 cm Area = ( 7 )2

= 7 7

= 154 cm2

Example: Diameter = 7 cm

Radius = 3 cm

Area = (3 )2

=

= cm2


Area of a circle = r2

rO

14

O

21

O

31

2OO

10O O

r

O

A

B

Exercise A

1. Radius = 14 cm Area =

Exercise B

1. Diameter = 21 cm Radius =

Area =

2. Radius = 3 cm

Area =

2. Diameter = 140 mm Radius =

Area=

3. Radius = 10 cm Area =

3. Diameter = 35 cmRadius =

Area =

3.5. AREA OF SECTOR


Area of minor sector AOB

=

7

120C

A

B

Example: Using =

Radius = 7 cm Area of minor sector AOB

= x 72

= x 7 x 7

= 38.5 cm 2

Example: Using = 3.142

Radius = 12 cm Area of minor sector

= 3.142 x 12 x 12

= 100.544cm2

Exercise 1. Radius = Area of minor sector AOB =

Exercise B

1. Radius = Area of minor sector AOB =

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

12 80

O

B

A

3.6 AREA OF SHADED REGION

Example :

Exercise :1.

2.

3.


7 7 7 7B7

A7 O

14Area of shaded region = - 2

= = 308

7

33

22

2

2

Area of shaded region =

=

10

O

40o

Area of shaded region =

=

Area of shaded region = -

=

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 , calculate

a) the perimeter of the whole diagram,b) the area of the shaded region. [ 6 marks]

Answer :

a) Perimeter of the whole diagram

=

= 58 cm

b) Area of the shaded region =

= cm2

Exercise1. In Diagram 1, JKL is arc of circle with centre M. NML is a straight line and JN = NM =

7 cm.

Using , 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 , 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. Using , 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

O F

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 , 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 , calculate

a) ,b) area of the shaded sector OTU,c) perimeter of the whole diagram.

[6 marks]


Diagram 5

QP O

S

R

T

U7 cm

Diagram 4

P

Q

R

S

T

UO 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 = PR and PR = 21 cm. Using , 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 , 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

TS

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 , 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 , 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.

Using , calculate

a) the arc of KLMb) the area of minor sector MONc) the area of the shaded region.

[6 marks]


P

Q

R

V

U

O T

S

110o

R

Diagram 8

Diagram 9

L

N

OM

K

60o

L

NO MK

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 , 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 , calculate

a) the area , in cm2 of the shaded regionb) the perimeter in cm, of the whole diagram. [7 marks ]


P O R L

S

K

Q

60o

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 , 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 , calculate

a) the perimeter, in cm, of the whole diagram,b) the area, in cm2, of the shaded region. [6 marks]


P

R

LS

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 60o.

Using , 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 , calculate

a) the area, in cm2, of the shaded regionb) the perimeter in cm, of the shaded region. [6 marks]

7. November 2006Length of Arc & Area of sector 16

P

R

120o

S

Q

45o

U

TDiagram 6

Diagram 5R

WU

S

W

T

O V

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.

OM = MP = 7 cm and .

Using 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 ] , 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

MP

60o

Diagram 5

Q

O

SR

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 ] , 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 ] calculate

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

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


60o

OS

R

Q

T

P

Diagram 3

O

B

FA C30o

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 .

Using 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 A1 88 2

cm3

cm

Exercise B1 44 2 440 3 110Exercise 3.2 Exercise A1 11 cm 2 49.5 cm 3 5.5.cm Exercise B1 6.982 cm 2 41.89 cm 3 9.426 cmExercise 3.3 Exercise A1 86 2

cm

Exercise B1 33.71 2 77.60Exercise 3.4 Exercise A1 616 2

cm23

cm2

Exercise B1 346.5 cm2 2 15400 mm2 3 962.5 cm2

Exercise 3.5 Exercise A1 462 cm2 2

cm2

Exercise B1 70.695 cm2 2 288.02 cm2

Exercise 3.61

cm22

cm23

cm2

Questions According to Examination Format

1a 115.5 cm2 1b 47.23 cm2a 50 cm 2b 115.5 cm2

3a 171.11 cm2 3b 109.78 cm4a 72 cm 4b 280 cm2

5a 450 5b 57.75 cm2 5c 55.5 cm6a 66 cm 6b 269.5 cm2

7a cm

7b cm2

8a cm

8b cm2

9a 152 cm 9b 981.75 cm2


10a cm

10b cm2

10c 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

cm b 154 cm2

SPM 2006J a 385 cm2 b 97 cm

SPM 2006 a cm

b cm2

SPM 2007J

a cm

b cm2

SPM 2007

a 243.83 cm2 b 104.33

SPM 2008J

a 83 b

2

1192

SPM 2008

A 64 b


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Chapter 3 II Arc Length n Sector Area ENHANCE

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