Circles. Circumferences of Circles diameter (d) O circumference (C) The circumference (C) and the...

Circles

Circumferences of Circles

diameter (d)

O

circumference (C)

The circumference (C) and the diameter (d) of a circle are related by

dC π

radius (r)

Since d = 2r, where r denotes the radius of the circle, we have

rC π 2

circumference of the circle

= 3.14 18 cm

= 56.52 cm

C = d

Do you know how to find the circumference of the circle with diameter 18 cm?

Take = 3.14,

Follow-up question 1

Complete the following table.

.

7

22 Take π

Radius Diameter

cm 14

m 7

mm 44

28 cm 88 cm

3.5 m 22 m

7 mm 14 mm

Circumference of the circle

d = 2r

C = d

Example 1

Solution

Find the circumference of a circle with diameter 15 m.

(Take = 3.14.)

Circumference of the circle

m 47.1

m 1514.3

m 15

Example 2

Solution

Find the radius of a circle with circumference 88 cm.

.

7

22 Take

Let r cm be the radius of the circle.

14

7

22 288

288

r

r

r

∴ The radius of the circle is 14 cm.

Example 3

Solution

The figure consists of a semi-circle and a rectangle from

which another semi-circle is cut from it.

Find the perimeter of the figure.(Give your answer correct

to 2 decimal places.)

The figure consists of a semi-circle and a rectangle from which

another semi-circle is cut from it. Find the perimeter of the figure.

(Give your answer correct to 2 decimal places.)

Perimeter of the larger semi-circle

cm 5

cm 102

1

Perimeter of the smaller semi-circle

cm 3

cm 62

1

∴ Perimeter of the figure

d.p.) 2 to(cor. cm 41.13

cm 16)(8

cm )10635(

Example 4

Solution

As shown in the figure, Karen’s antique bicycle has two

wheels of radii 35 cm and 15 cm respectively. If the larger

wheel makes 18 revolutions, find

(a) the distance travelled by the bicycle,

(b) the number of revolutions that the smaller wheel has

made.

.

7

22 Take






made.

.

7

22 Take

(a) The required distance travelled by the bicycle

cm 3960

cm 18357

222

cm 18)352(






made.

.

7

22 Take






made.

.

7

22 Take

(a) The required distance travelled by the bicycle

cm 3960

cm 18357

222

cm 18)352(

(b) Circumference of the smaller wheel

fig.) sig. 3 to(cor.cm 7

660

cm 157

222

cm 152

∴ The number of revolutions that the smaller wheel

has made

427

6603960



660

cm 157

222

cm 152


has made

427

6603960



660

cm 157

222

cm 152


has made

427

6603960



660

cm 157

222

cm 152


has made

427

6603960



660

cm 157

222

cm 152


has made

427

6603960

Areas of Circles

1. Prepare a circle.

2. Divide it into 16 equal parts.3. Cut the circle into 2 parts as shown.4. Rearrange them to form the following figure.

The formula of the area of circle can be deduced in the following way:

Which kind of quadrilateral does the figure look like? parallelogram

Assuming the radius of the circle is r, estimate the height and the base of the figure obtained in terms of r.

height

base

r

Height r Base2

1 2 r = r

The figure looks like a parallelogram.

∴ The approximate area height base

If A and r denote the area and the radius of a circle respectively, then

r2

rA π

= 282 cm2

7

22

For a circle of radius 28 cm,

its area

= 2464 cm2

A = r 2

take = , 7

22


Complete the following table. (Take = 3.14.)

Radius Diameter circle the of Area

mm 4

cm 102m 40

(Give your answers correct to 2 decimal places if necessary.)

8 mm 50.27 mm2

5 cm 78.54 cm2

3.57 m 7.14 m

d = 2r

A = r2

Example 5

Solution

(a) Find the area of a circle with radius 6 cm.

(b) Find the radius of a circle with area 40 cm2. (Give

your answers correct to 3 decimal places.)







(a) Area of the circle

d.p.) 3 to(cor.cm 097.113

cm 62

22

(b) Let r cm be the radius of the circle.

d.p.) 3 to(cor.568.3

40

40

40

2

2

r

r

r

∴ The radius of the circle is 3.568 cm.

Example 6

Solution

In the figure, two semi-circles are cut from a square ABCD.

Find the shaded area correct to 2 decimal places.

In the figure, two semi-circles are cut from a square ABCD.

Find the shaded area correct to 2 decimal places.

Radius of the two semi-circles cm 2 cm 24

∴ Area of the two semi-circles 222 cm 4cm 22

1)2(

Area of the square 222 cm 64cm)44( ABCD

∴ Area of the shaded area area of the square ABCD area of the two semi-circles.

d.p.) 2 to(cor.cm 43.51

cm )464(2

2

Example 7

Solution

The figure consists of 3 semi-circles having the same

centre with diameters 6 cm, 8 cm and 10 cm respectively.

Find the area of the shaded region in terms of .

The figure consists of 3 semi-circles having the same

centre with diameters 6 cm, 8 cm and 10 cm respectively.

Find the area of the shaded region in terms of .

Area of the semi-circle with diameter

2

22

cm 5.12

cm2

10

2

1cm 10


2

22

cm 8

cm2

8

2

1cm 8


2

22

cm 5.4

cm2

6

2

1cm 6

∴ Area of the shaded region 2

2

cm 9

cm )5.485.12(

Example 8

Solution

The area of a semi-circle is 50 cm2. Find the perimeter of the

semi-circle, correct to 2 decimal places.

Let r cm be the radius of the semi-circle.

100

100

502

1

2

2

r

r

r

∴ The perimeter of the semi-circle

d.p.) 2 to(cor.cm 29.01

cm 100

22

11002

Arcs and Sectors

To avoid confusion,

the shorter arc ACB

Arcs

If A and B are two points on the circumference of a circle,

then curve AB is called an arc,

The figure shows a circle with centre O.

A

B

O

ABwhich is denoted by ‘arc AB’ or ‘ ’.

and the longer arc ADB

C

D

AOB is the angle subtended at the centre by the arc AB.

It can be simply called the angle at the centre.

angle at the centre

is denoted by

ACBis denoted by .

ADB

Lengths of Arcs

Let be an arc on the circle AB

For a circle with centre O and radius r,

O

rcircumference = 2 r.A

B

and be the angle subtended at the centre

by the arc.

Let’s find the lengths of arc AB for different values of ...

Length of

AB O

180

A

B

(a) = 180

= 360

180

of the circumference circumference 2

1

= 2 r 2

1 r

2 r 8

1 r

4

1

= 360

4

5

of the circumference

circumference 4

1

Length of

AB

(b) = 90

= 360

90

of the circumference

= 2 r 4

1 r

2

1

O

90

A

B

circumference 8

1

Length of

AB

=

(c) = 45

O

45

AB

= 2 r 360

1

180

1 r

360

1

of the circumference circumference 360

1

Length of

AB

(d) = 1

= O

1A B

What can you conclude from the above results?

The ratio of the arc length to the

circumference of the circle = . 360

Therefore,

O

r

A

BFor a circle with radius r and angle subtended at the centre by an arc,

rθ

2360

π

length of arc =

Length of arcCircumference

Angle subtended at the centreRound angle=

Follow-up question 3In each of the following figures, O is the centre of the circle.

O72

A

B

10 cm

O

240

A

B

9 m

Solution

cm 4

cm 102360

72 1.

π

π

AB

m 12

m 92360

240 2.

π

π

AB

Find the length of the arc AB in terms of .

1. 2.

Example 9

Solution

In the figure, O is the centre of the circle. Find the length of arc ABC

correct to 2 decimal places.

In the figure, O is the centre of the circle. Find the length of arc ABC

correct to 2 decimal places.

The length of arc

d.p.) 2 to(cor.cm 22.83

cm 62360

218

cm 62360

142360

ABC

Example 10

Solution

The figure shows a circle with radius 15 cm. If , find .

The figure shows a circle with radius 15 cm. If , find .

∵ cm 6

61

ABC

∴

238

12236030

360

6

61360

6

61152

360

360

The figure shows a circle with radius 15 cm. If cm6

61

ABC , find .

Example 11

In the figure, ∠ABC is the angle subtended at the centre C by arc AB.

BDC is a semi-circle with diameter BC. If AC 10 cm and ∠ACB 70 ,

find the perimeter of the figure correct to 3 significant figures.

In the figure, ∠ACB is the angle subtended at the centre C by arc AB.

BDC is a semi-circle with diameter BC. If AC 10 cm and ∠ACB 70 ,

find the perimeter of the figure correct to 3 significant figures.

SolutionBC AC 10 cm (radii)

cm 5

cm 102

1

CDB

cm 9

53

cm 102360

70

AB

Perimeter of the figure

fig.) sig. 3 to(cor.cm 9.37

cm 109

355

ACABCDB

Areas of Sectors

O

A

B

A sector is the region enclosed by an arc and two radii of a circle.

In the figure, AOB is a sector enclosed

by , radii OA and OB.

AB

AOB is called the angle of the sector. angle of the sector

sector

For a sector with radius r and angle of the sector ,

angle roundsector the of angle

circle of area

sector of area

BO

A

r θ

360θ

π 2r

sector of area

Therefore,

2

360r

θ π

area of sector =

Refer to the following figure.

O

AB

324

10 m

22 m 10360

324

πArea of sector OAB

2m 90π2

360

sector of area

rθ

π

Radius sector the of Angle sector of Area

mm 10

cm 8


Complete the following table.)necessary. if of terms in answers your(Give π

10 mm2

3 m

8 cm2

8 m2

36

2

360

sector of area

rθ

π

320 45

Example 12In the figure, AOB and COD are sectors. Given that BD 4 cm, OD 7 cm

and ∠AOB 130 , find the area of the shaded region ABDC correct to 2

decimal places.

In the figure, AOB and COD are sectors. Given that BD 4 cm, OD 7 cm

and ∠AOB 130 , find the area of the shaded region ABDC correct to 2

decimal places.

SolutionArea of sector

2

22

cm 36

1573

cm )47(360

130

AOB

Area of sector

2

22

cm 36

637

cm 7360

130

COD

∴ Area of the shaded region

d.p.) 2 to(cor.cm 68.81

cm 26

cm 36

637

36

1573

2

2

2

Example 13

Solution

It is given that the radius of a pizza is 12 cm. As shown in the figure,

one slice of the pizza with area 24 cm2 is eaten. Find the angle of

the remaining sector of the pizza.

It is given that the radius of a pizza is 12 cm. As shown in the figure,

one slice of the pizza with area 24 cm2 is eaten. Find the angle of

the remaining sector of the pizza.

It is given that the radius of a pizza is 12 cm. As shown in

the figure, one slice of the pizza with area 24 cm2 is eaten.

Find the angle of the remaining sector of the pizza.

∵ Area of the sector 24 cm2

∴

300

60360144

36024360

2412360

360 2

Example 14

Solution

In the figure, the area of sector AOB is 80 cm2 and ∠ AOB 120 . Find the radius of the sector, correct to 3 significant figures.

In the figure, the area of sector AOB is 80 cm2 and ∠ AOB 120 . Find the radius of the sector, correct to 3 significant figures.

Let r cm be the radius of the sector. ∵ Area of the sector 80 cm2

∴

fig.) sig. 3 to(cor.18.6240

36080

80360

240

80360

120360

2

2

r

r

r

∴ The base radius of the sector is 6.18 cm.

Cylinders

A cylinder is a solid with uniform cross-section and its two bases are circles.

The solids below are all cylinders.

Volume of CylindersWe have learnt that

Similarly,

r

h

base

For a cylinder of base radius r, height h and volume V,

then

volume of a cylinder = base area × height

volume of a prism = base area × height

V = r 2 h


5 cm

2 cm

Volume of the cylinder

= 22 5 cm3

= 20 cm3

= 62.83 cm3 (cor. to 2 d.p.)

V = r 2 h


The figure shows a cylinder of base radius r and height h. Complete the following table.

)necessary. if of terms in answers your(Give π

(cm) r (cm) h )(cm Volume 3

4 7

2

11r

h112

10

5

40

275

V = r2 h

Example 15

Solution

The figure shows a dumbbell which consists of three cylinders of

length 10 cm each. The base radii of the left and the right cylinders

are 4 cm and the base radius of the middle cylinder is 1 cm. Find the

volume of the dumbbell. (Give your answer in terms of)

Volume of the dumbbell

3

3

322

cm 330

cm )10320(

cm ]1012)104[(

The figure shows a dumbbell which consists of three cylinders of length

10 cm each. The base radii of the left and the right cylinders are 4 cm and

the base radius of the middle cylinder is 1 cm. Find the volume of the

dumbbell. (Give your answer in terms of .)

Example 16

Solution

900 cm3 of water is poured into a cylindrical tank and the depth of water is 12 cm. Find

the base radius of the tank, correct to 3 significant figures.

Let r cm be the base radius of the tank.

fig.) sig. 3 to(cor.89.412

900

900122

r

r

∴ The radius of the tank is 4.89 cm.

Example 17

Solution

The figure shows the metal cup whose inner and outer diameters are 4

cm and 6 cm respectively. If the height and the thickness of the base

of the cup are 7 cm and 1 cm respectively, find the volume of metal

required to make the metal cup.(Give your answer in terms of .)





Volume of metal required

3

3

322

cm 39

cm )2463(

cm )17(2

47

2

6









Example 18

Solution

In the figure, water flows from a pipe into a cylindrical tank of base

radius 0.8 m and height 1.2 m. In the beginning, the tank is half filled

with water. If the water flows from the pipe at a constant rate of 50

cm3/s, find the time taken to fill up the tank in minutes.

In the figure, water flows from a pipe into a cylindrical tank of base

radius 0.8 m and height 1.2 m. In the beginning, the tank is half filled

with water. If the water flows from the pipe at a constant rate of 50

cm3/s, find the time taken to fill up the tank in minutes.

The unfilled volume of the tank

3

3

32

cm 000 384

m 384.0

m 2

12.18.0

The time taken to fill up the tank

minutes 128

s 7680s/cm 50

cm 000 384

flow water of rate

tank theof volumeunfilled

3

3

Total Surface Areas of Cylinders

1. Prepare a cylindrical can that has a piece of wrapper.

2. Cut the wrapper vertically as shown.

3. Spread out the wrapper.

The lateral face of a cylinder is a curved surface. How can we find the curved surface area?

A

B C

D

What is the relationship between AD and the circumference of the base of the can?

AD = circumference of the base of the can

What is the relationship between AB and the height of the can?

AB = height of the can

Then, what is the curved surface area of the can?

A

B C

D

Curved surface area of the can

= area of rectangle ABCD

= AD × AB

= the circumference of the base of the can

× the height of the can

For a cylinder of base radius r and height h,

The total surface area of the cylinder can be found by adding the areas of its two bases to its curved surface area:

curved surface area of a cylinder = 2 r h

base area = r 2

r

curved surface area = 2rh

base area = r 2

total surface area of a cylinder = 2 r h + 2 r 2


3 cm

4 cm

Total surface area of the cylinder

Total surface area = 2 r h + 2 r 2

22

cm 2

4234

ππ=

= 20 cm2

(cm) r (cm) h )(cm area surface Total 2

1 3

5 π120

42


The figure shows a cylinder of base radius r and height h. Complete the following table.

)necessary. if of terms in answers your(Give π

r

h

7

Total surface area = 2 r h + 2 r

2

Example 19(a) Find the total surface area of the cylinder in the figure.

(b) The solid as shown is half of the cylinder in (a). Find the total surface area of

the solid.

(Give your answers correct to the nearest cm2.)

(a) Find the total surface area of the cylinder in the figure.


the solid.




the solid.




the solid.


(a) Total surface area of the cylinder

)cmnearest the to(cor.cm 101

cm 32

cm )824(

cm 2

426

2

42

22

2

2

22

(b) Total surface area of the solid

)cmnearest the to(cor.cm 74

cm )1624(

cm 2

13246

22

2

2



the solid.




the solid.


Solution

Example 20

Solution

If the curved surface area and the height of a cylinder are 42 cm2

and 7 cm respectively, find its volume in terms of .

Let r cm be the base radius of the cylinder. ∵ Curved surface area 42 cm2

∴

314

42

4272

r

r

∴ Volume of the cylinder 3

32

cm 63

cm 73

Extra Teaching Example

Example 2 (Extra)

Solution

The figure shows a ring formed by two circles with a common centre

O. Find the perimeter of the ring.

(Take = 3.14.)

The figure shows a ring formed by two circles with a common centre

O. Find the perimeter of the ring.

(Take = 3.14.)

Circumference of the inner circle

cm 68.37

cm )6( )14.3(2

Circumference of the outer circle

m 8.62

cm 0)1( )14.3(2

cm 4)(6 )14.3(2

∴ The perimeter of the ring

cm 100.48

cm 62.8 cm 68.37

Example 8 (Extra)

Solution

In the figure, ABCD is a square and ADE is a semi-circle of radius r

cm. The area of ABCDE is 40 cm2.

(a) Find r.

(b) Find the perimeter of ABCDE. (Give your answers correct to 2

decimal places.)



(a) Find r.


decimal places.)

(a)

d.p.) 2 to(cor.68.2

...6796.22

14

40

402

14

402

14

402

1)2(

2

22

22

r

r

rr

rr

)(68.22

14

40

402

14

402

14

402

1)2(

2

22

22

準確至二位小數

r

r

rr

rr

)(68.22

14

40

402

14

402

14

402

1)2(

2

22

22


r

r

rr

rr(a)

)(68.22

14

40

402

14

402

14

402

1)2(

2

22

22


r

r

rr

rr



(a) Find r.


decimal places.)

Solution

(b) Perimeter of

d.p.) 2 to(cor.cm 24.50

cm (2.6796)] [6(2.6796)

cm )6(

cm )22

123(

rr

rrABCDE

Example 10 (Extra)

Solution

The figure shows a circle with radius 18 cm. If 18

ABC cm, show

that is a semi-circle.

The figure shows a circle with radius 18 cm. If cm, show

that is a semi-circle.

The figure shows a circle with radius 18 cm. If cm, show

that

ABC is a semi-circle.

∵ cm 18ABC

∴

18036

36018

18182360

AOC

AOC

∴ ABC is a semi-circle.

Example 14 (Extra)In the figure, the perimeter of sector AOB is 12 cm and ∠ AOB =

140 . Find the radius and the area of sector AOB.

.

7

22 Take

In the figure, the perimeter of sector AOB is 12 cm and ∠ AOB =

140 . Find the radius and the area of sector AOB.

.

7

22 Take

SolutionLet r cm be the radius of the sector. ∵ Perimeter of sector AOB 12 cm

∴

7.2

1227

222

360

140

1222360

140

r

r

rr

∴ The radius of the sector is 2.7 cm.

Area of sector

2

22

22

cm 91.8

cm 7.27

22

360

140

cm 360

140

r

Example 19 (Extra)The figure shows a solid formed by drilling a cylindrical hole of base

radius 6 cm from a rectangular wooden block of dimensions 20 cm

30 cm 5 cm. Find the total surface area of the solid. (Give your

answer correct to 1 decimal place.)

The figure shows a solid formed by drilling a cylindrical hole of base

radius 6 cm from a rectangular wooden block of dimensions 20 cm

30 cm 5 cm. Find the total surface area of the solid. (Give your

answer correct to 1 decimal place.)

Solution

Total surface area of the solid

d.p.) 1 to(cor.cm 3.1662

cm )72602003001200(

cm )625622520253023020(

2

2

22

Circles. Circumferences of Circles diameter (d) O circumference (C) The circumference (C) and the...

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