chp 10 booklet

7/27/2019 Chp 10 Booklet

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CHAPTER 10

CIRCLES


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(A) To Identify Parts of a Circle

Centre Circumference

Radius Diameter

Chord Arc

Sector Segment

(B) To Draw a Circle and Parts of a Circle

Draw the circles below:

1. Draw a circle with centerO and radius 2 cm.

2. Draw a circle given its diameter is 6cm.

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3. Draw a diameter of 3cm passing through a point Con a circle

with centre O.

4. Draw two chords of length 4cm passing through a pointEon the

circumference of the circle with radius 3cm.

5. Draw the sector of a circle given that the angle at the centre is

60o and its radius is 4cm.

3


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(C) To Determine The Centre Or Radius Of A Circle By

Construction

The perpendicular bisector of chords of a circle intersect at thecentre of the circle.

1. Determine the centre and radius of the circles given below:

(a) (b)

(D) To Understand and Use the Concept of Circumference to Solve

Problems.

Formula : Circumference = 2r

4

==

22

73.142


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(i) To find the circumference of a circle given its diameter or

radius

1. Find the circumference of a circle of

(a) diameter 14 cm (b) radius 2.1 cm

2. Take = 3.142, find the circumference of the circles below:

(a) (b)

3. CirclePhas a radius of 21cm and circle Q has a diameter of5.6cm. Find the difference in the circumference of circlePand

Q.

(Take =7

22)

5

O

3 cm

O

10 cm


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(ii) To find the diameter or radius of a circle given its

circumference

1. Take =722 , find the diameter of the circle with circumference

17.6cm.

2. Take = 3.142, find the radius of a circle given its

circumference is 314.2 cm.

3. A circle and a square with length 5 cm have the same perimeter.Find the radius of the circle, correct to 2 decimal places.

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(iii) To solve problems involving the circumference of circles.

1. A circle is inscribed inside a square of length 1.4 cm as shown.

Find

(a) the circumference of the circle.

(b) the difference in the perimeter of the

square and the circle.

2. The tyre of a bicycle rotates 50 times per minutes. Given the

diameter of the tyre is 42 cm, find the distance traveled by the

bicycle in 5 minutes. Give your answer in meters.

3. XYZis a right-angled triangle. Semicircles are drawn on all 3

sides of the triangle. Find the perimeter of the diagram.

(E) Arc of a Circle

7

1.4cm

5 cm

13 cm

cm

Z Y

X


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The length of arc is proportional to the angle at the centre

Formula:

=

360

CentreAtAngle

nceCircumfere

arcofLength

(i) To find the length of the arc given the angle at the centre and

the radius

1. Find the length of the arc of a circle with radius 4.9cm and anangle of 56o at the centre.

2. Find the value ofx for each of the following circles:(a) (b)

3. The figure shows a circle of diameter 12.6cm and and arcAB

that subtends an angle of 80o at the centre. Find the length of arc

AB.

8

Length of Arc

=

Angle at centre

Circumference 360o

O

7 cm

72o x

O3.5cm

cm

300o

x

O80o

A

B


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(ii) To find the angle at the centre given the length of the arcand the radius of the circle

1. Find the angle subtended at the centre of the circle, given

(a) radius = 35mm, length of arc = 143mm(b) radius = 28cm, length of arc = 22cm

(Take =7

22)

2. Find the angle subtended at the centre of a circle by an arc of

length 12.1 cm, given that its radius is 3.5cm. (Take =

7

22)

3. Find the value ofx given that the circle with centre O has aradius of 14m and the minor arcAB is 44m long.

9

O

xAB


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(iii) To find the radius of a circle given the length of thearc and the angle at the centre

1. Find the radius of the circle, given

(a) angle at the centre = 36o, length of arc = 8.8cm(b) angle at the centre = 45o, length of arc = 121mm

(Take =7

22)

2. Find the radius of a circle given that an arc of length 19.8m

subtends an angle of 180o at the centre of the circle.

3. Find the radius of the circle with centre O given that the majorarcPQ is 121cm.

10

O30o

P

Q


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(iv) To solve problems involving arcs of a circle

4. The minute hand of a clock has traveled one quarter ofa revolution. Find the distance traveled by the tip of theminute hand given that the minute hand is 12.1cm.

5. The hour hand of a clock moves from 12am to 8am, covering a

length of7

250 cm. Find the length of the hour hand.

6. The figure shows a right-angled triangle joined to a semicircle.

find the perimeter of the figure.

11

6cm

8cm


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7. The figure given shows a rectanglePQRSoverlapped by twoquadrantsPAD andRCB of radius 7cm. IfPS= 10cm andPQ

= 14cm, calculate the perimeter of the shaded area.

(F) Area of a Circle

Formula:

(i) To find the area of a circle given the radius or diameter

1. Take =7

22, find the area of a circle with radius 7 cm.

2. Take =3.14, find the area of a circle with radius 5.6 cm.

3. Take =7

22, find the area of a circle with diameter 42cm.

(ii) To find the radius or diameter given the area of a circle

12

P A S

B

RC

Q

D

Area of circle = r2


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1. The area of a circle is 154 cm2. Find its radius and diameter.

(Take =7

22)

2. A circle has an area of 78.55cm2. Determine the diameter of the

circle given =3.142.

3. The circumference of a circle is 88cm. Calculate the area of the

circle.

4. Determine the circumference of a circle which has an area of616cm2.

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5. The figure shows two concentric circles with centre O. Find thearea of the annulus.

(iii) To solve problems involving area of circles

1. Take = 7

22

, find the area of the shaded regions below.

(a) (b)

14

O

3 cm4 cm

Important!

ConcentricCircles= circles havingthe same centre

Annulus= area betweentwo concentriccircles

O

14cm

24cm

18cm

O P

42cm


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2. The diameter of the circle has the same length as the sides of the

square. Take =3.142, find the area of the figure below.

3. Mr Lingam bought a pizza which is circular in shape. His

daughter ate quarter of the cake. Given the area of the topsurface of the remaining portion is 462 cm2, find the diameter of

the pizza.

15

O

28 cm


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(G) Area of a Sector of a Circle

Formula:

(i) To find the area of a sector given the radius and angle at the

centre

1. Find the area of the sector of a circle, given

(a) Radius = 14cm (b) Radius = 24m

Angle at centre = 45o Angle subtended at

centre = 210o

(Take =7

22)

2. Find the area of each of the shaded regions where O is the centre

of the circle.(a) (b)

16

Area of

Sector

= Angle at centre

Area of Circle 360o

1 circlesemicircle

= circle

A quarter of a circle

= circle

O

72o

7cmO

72o

7cm 4.9mO

=

360

CentreAtAngle

CircleofArea

SectorofArea


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(ii) To find the angle at the centre given the radius and area of a

sector

1. Find the angle at the centre for the circles shown below.

(a) (b)

2. The area of the sector of a circle, with the radius 4.2cm, is

15.28cm2

. Find the angle at the centre.

17

O 21 m

Area of sector = 77 m2

2.8cm

O18.75cm2


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(iii) To find the radius given the area of a sector and the angle at the

centre

1. The area of the sector of a circle, with an angle of 70o at the

centre is 22 m2. Find the radius of the circle. (Take =7

22)

2. The sector of a circle subtends an angle of 60o at the centre of

the circle. Given the area of the sector is 231cm2, find thediameter of the circle.

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(iv) To solve problems involving area of sectors and area of circles

1. The diagram below shows a circle with centre O and a radius of

14mm. OAB is a right-angled triangle. Find the area of the

shaded part.

2. Find the area of the shaded part for the diagram below. (Take

=7

22)

19

36o

6cm10cm

O

A

B


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PMR Past Year Questions (Chapter 10 Circles)

2004 Paper 1 Question 19

Diagram 12 shows a circle with centre O.

The radius of the circle is2

110 cm. Calculate the length, in cm, of the

minor arcMN. (Use7

22= )

A2

55B

2

77C

8

1155D

81617


In Diagram 13,POR is the diameter of circlePQR.

Given thatPQ = QR andPR = 14cm, calculate the area, in cm2

, of theshaded region. (Use

7

22= )

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A 56 B 93 C 105 D 142


In Diagram 25,PQR is an arc of a circle with centre O andPRO is an

equilateral triangle.

The perimeter, in cm, of the whole diagram is

(Use7

22= )

A 42 B 63 C 64 D 85

2005 Paper 1 Question 23In Diagram 18,PQR is a straight line andPQSis a quadrant of a circle

with centre Q.

The area if triangle QRSis 21cm2. Calculate the area, in cm2, of

quadrantPQS. (Use7

22= )

A 11 B 19.25 C 38.5 D 154

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In Diagram 19,PQR is an arc of a circle with centre O.

The radius of the circle is 14 cm andPT= 2 TS. Calculate the perimeter,

in cm, of the whole diagram. (Use7

22= )

A 66 B 94 C 98 D 126


Diagram 25 shows a circle with centre O. The length of the minor arc

PQ is 3.3 cm.

Calculate the radius, in cm, of the circle. (Use7

22= )

A 2.42 B 4.50 C 14.85 D 29.70

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Calculate the length, in cm, of the minor arcPQ. (Use7

22= )

A 22 B 44 C 88 D 176

2006 Paper 1 Question 21Diagram 16 shows the surface of a tablePQRSTU.PQSTis a rectangle.

PUTand QRSare semicircles with centresXand Yrespectively.

It is given thatPT= 1.4 m andPQ = 3 m. Calculate the perimeter, in m,

of the surface of the table. (Use7

22= )

23


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A 10.4 B 13.2 C 14.8 D 17.6


Diagram 17 shows two sectors, QSR and UST, with the common centre

S.RSUand QSTare straight lines.

It is given that ST= 2QS. Calculate the area, in cm2, of the sectorUST.

(Use7

22= )

A 11 B 19.25 C 38.50 D 77


Diagram 16 shows a circle with centre O.PT, QU,RVand SWare

diameters of the circle.

Which of the following minor arcs is the longest?A TU B UV C WP D RS

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Diagram 17 shows a circle with centre O and radius 6 cm.

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

A 18 B 5.4 C 10.2 D 30.6


Diagram 22 shows a square OPQR and an arcRSPwith centre O.

Calculate the perimeter, in cm, of the whole diagram.A 61 B 58 C 47 D 33

25


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Calculate the area, in cm2, of the coloured region. (Use7

22= )

A 105 B 116 C 539 D 616


Diagram 13, O is the centre of the circle andPOR is a diameter of the

circle.PQRSis a rectangle.

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It is given thatPR = 10 cm andPQ = 6 cm. Find the area, in cm2, of thecoloured region.

A 25 48 B 25 96 C 100 48 D 100 96


In Diagram 19,PQRSis a square and STUis an arc of a circle with

centreP.

The area ofPQRSis 576 cm2. Calculate the length, in cm, of the arc

STU.

A 16 B 28 C 32 D 48

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chp 10 booklet

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