Area and Volume (F.1)

Figure (a)

F.1 Mathematics Supplementary Notes Chapter 11&12 5/2002 P. 1

Chapter 11 Estimation in Measurement Name:___________( )Class: F.1 ( )

Chapter 12 Area and Volume (1)

Important Terms

degree of accuracy 準確度 maximum error 最大誤差 cuboid 長方體actual value 真確值 accumulated error 累積誤差 surface area 表面積measured value 量度值 trapezium 梯形 uniform cross-section 均勻橫切面approximate 近似 solid 立體 prism 棱柱error 誤差 cube 正方體 lateral face 側面

Revision Notes

1. The Approximate Nature of Measurement

(a) All measurements are approximations.

With the ruler in Figure (a), the

length of the wingspread( 翼幅 )

is between 5cm and 6cm.

Since it is closer to 6cm, we say the length is

approximately 6, and we write length 6cm

The measurement is correct to the nearest cm.

(準確至最接近的厘米)

With the ruler in Figure (b), the

length of the wingspread is between 5.7cm and 5.8cm.

Since it is closer to 5.8cm, we say the length is

approximately 5.8cm, we write length 5.8cm

The measurement is correct to the nearest 0.1cm.

(準確至最接近的 0.1厘米) OR

correct to the nearest mm.

(準確至最接近的毫米)

(b) The error of a measurement is the difference between the measured value and the actual value.

e.g. If the actual length of a car is 4.55m and it is measured as 4.4m, then the error of the measurement

= (4.55 – 4.4)m

=0.15m

Figure (b)


(c) The largest difference between the actual value and the measured value is called the maximum error of the

measurement and it is equal to difference between the finest markings on the measuring tool.

e.g. If the distance between the finest markings on a ruler is 1mm, then the maximum error of a measurement

by this ruler = 1mm = 0.5mm.

2. Measuring Tools and Units for Direct Measurement

When we measure a quantity, we have to

(a) choose an appropriate unit so that other people can understand the result easily;

(b) choose an appropriate measuring tool to achieve the desired degree of accuracy.

Commonly used units for measurement:

For length : mm , cm , m and km

For area : mm2, cm2, m2 and km2

For volume: mm3, cm3, m3 and km3

For weight: kg , g and mg

For time : h , min and s

Example

1. Convert the area 123 400 cm2 into m2 .

Solution 123 400 cm2 = 123 400

= 12.34 m2

2. Convert the volume 123 400 cm3 into m3 .

Solution 123 400 cm3 = 123 400

= 0.1234 m3

3. Area of Simple Figures

Choose the most appropriate(合適 ) units so that people can understand the meaning easily.

1m = 100cm 1m3 = 1 000 000cm3

i.e. 1cm3 = m3

1m = 100cm 1m2 = 10 000cm2

i.e. 1cm2 = m2


Rectangle (長方形, 矩形) Parallelogram (平行四邊形)

Let the length ( 長 ) be l

the breadth ( 闊 ) be b b

Area of rectangle = length breadth

= l b

Let the base ( 底 ) be b

the height (高) be h

Area of parallelogram = base height

= b h

Triangle (三角形) Trapezium (梯形)

Let the base ( 底 ) be b


Area of triangle = base height

= b h

Let the upper base be a

the lower base be b


Area of trapezium

= (upper base lower base) height

= (a b) h

Solids with Uniform Cross-Section

Cuboid (長方體, 矩體)

( each face is a rectangle )

Cube (正方體)

( each face is a square)

Let the length be l

the breadth be b

Let the length = the breadth = height = h

b

l

b

l

hh

h

b b

b

altitude

a

b

h

b bb

h hh

lb

h

h

h

h


the height be h

Volume of the cuboid

= length breadth height

= l b h

Volume of the cube

= length breadth height

= h3

4. Solids with Uniform Cross-sections

Shade the uniform cross-sections of each of the following solids.

Volume of solid with uniform cross-section

The cross-sections(橫切面 ) have the same shape and size

Solid with uniform cross-section

(有均勻橫切面立體 )


= area of cross-section length

Example. A cuboid has a square base of area 16cm2. If the volume of the cuboid is 180cm3,

find its dimensions.

Solution Let the length = the breadth = l cm, the height = h cm

base area of the cuboid l 2 = 16

l = 4

volume of the cuboid l l h = 180

16 h = 180

h = 11.25

the dimensions of the cuboid are 4cm 4cm 11.25cm

Exercise 1: Choose the most appropriate unit for each of the following

Answer

1. The time for 100 m dash( 短跑 ) __________

2. The area of this classroom. __________

3. The distance from Hong Kong to Tokyo. __________

4. The weight of our body. ––––––––––

5. The thickness of a $1 coin. __________

6. The time taken to fly from Hong Kong to Shanghai on an aeroplane. __________

7. The weight of a letter. __________

8. The time taken to boil an egg. __________

9. The volume of a cup. __________

10. The height of Victory Peak. __________

11. The area of Hong Kong. __________

12. The volume of water in a swimming pool. __________

Exercise 2

1. The approximate volume of water in

the measuring cylinder( 量筒 ) is __________

The measurement is correct to the nearest __________

l cm

l cm

h cm

Dimensions are written in the form length breadth height


2. The approximate size of AOB is ___________ .

The measurement is correct to the nearest ___________.

3.

The approximate length of the stick is __________.

The measurement is correct to the nearest _________.

4.

The approximate weight of the watermelon is __________.

The measurement is correct to the nearest __________ .

5.

Measure the lengths of the sides of triangle ABC with a ruler

correct to the nearest 0.2cm.

A

B

C


AB = , BC = , AC =

Exercise 3

2. Convert the following areas into cm2.

(a) 0.00042km2 (b) 21m2

3. Convert the following areas into m2.

(a) 1 500 000 cm2 (b) 1.45 km2

4. Convert the following area into km2.

(a) 156 700 cm2 (b) 9870 m2

5. Convert the following volumes into cm3.

(a) 0.000 000 042km3 (b) 2.12m3

6. Convert the following volumes into m3.

(a) 1 500 000 cm3 (b) 2.345 km3

7. The following figure shows a triangle ABC, find

(a) the area of ABC,


(b) the value of h.

8. In the following figures, find the area of the shaded region.

(a)

(b)

9. A rectangle has its length twice its width. If the length is 16cm, find the area of the rectangle.

10. The area of a trapezium is 270cm2, the upper base is half of the lower base, the altitude is 60cm. Find the

lower base of the trapezium.

11. In the figure, area of the rectangle EGCD is 108cm2, DE is 9 cm. The areas of the two squares AHKE and

HBGK are equal. AED and EKG are straight lines. Find

(a) AE,

(b) the area of square HKGB.

A

KH

E D

CB G


12. ABCG and CDEF are two squares. BCD and FGC are straight lines.

AB = 10cm, ED = 14cm. Find the area of BGE.

Exercise 4

1. The figure shows a U-shaped solid with uniform cross-section (shaded).

If the area of the cross-section is 15cm2 and the length is 8cm,

find the volume of the solid.

Solution Volume of the solid = area of cross-section length

= ( ____ ____ )cm3

=

2. The figure shows a solid with uniform cross-section ABCDE.

The dimensions are in cm. Find

(a) the area of the cross-section ABCDE,

Solution area of ABE =

=

area of BCDE =

=

area of the cross-section ABCDE =

=

(b) the volume of the solid.

Solution length of the solid =

area of the cross-section =

volume of the solid =

=

3. The length and the breadth of a cuboid are both equal to 10cm, and its height is

twice its length. Find the volume of the cuboid.

A

F E

DCB

G

14cm10cm


4. The breadth of a cuboid is half its length, its altitude is three times its breadth.

Let x cm be the length of the cuboid.

(a) Express the breadth and the altitude of the cuboid in terms of x.

(b) If the altitude of the cuboid is 9cm, find the dimensions of the cuboid.

5. Find the volumes of the following solids with uniform cross-sections.

(a)

(b)

(c)

(d)


Level II

6. Find the volume of the open wooden box.

7. (a) In the figure, the dimensions of the tank are 40cm 20cm 50cm. 24000cm3 of water is poured into

a rectangular tank. Suppose the depth of water in the tank is h cm. Find the value of h.

(b) The length of a metal cube is 20cm, what is its volume?

(c) If the metal cube is put into the rectangular tank, does the water

overflow? Why?

8.

A piece of rectangular clay(黏土) with dimensions 10cm 5cm 2.5cm is pressed in form of a cube.

20 cm

40 cmh cm


What is the length of the cube?

9. If 18cm3 of water is poured into a bottle with uniform cross-section, the water level rises 3cm. What is the

area of the cross-section of the bottle?

10. If the cross-sectional area of each of the two bottom layers of the

wedding cake in the figure is double the cross-sectional area of the

layer on its top, and all the three layers have the same height,

find the volume of the cake.

11. The given net(摺紙圖樣) can be bolded to form a solid. Find

(a) the volume of the solid

(b) the total surface area of the solid

12. The figure shows a prism drawn on the isometric grids (等距方格). The length of each line segment on the

isometric grids represents 1 cm. Find

(a) the area of the uniform cross-section

1 cm 4 cm

2 cm


(b) the volume

13. In the following figures, find the area of the shaded region.

(a)

(b) ABFG and CDEF are two squares.

14. In the following figures, find the total surface area of the solids which are made from

cubes whose sides are 1 cm.

(a)

(b)

6cm

8cm

4cm

A B

C D

EFG

5cm3cm


Level III

15. A1 , A2 , A3 and A4 are the areas of the triangles.

If A1= A2 =A3 +A4 , find A4 .

(Ans:15cm2 )

16. Find x. (Ans: 37.5cm2)

17. Find the volume of the solid.

(Ans : 24cm2)

*18. ABCG and CDEF are two squares. BCD and FGC are straight lines.

AB = 8cm, Find the area of BGE.

(Ans: 32cm2)

9cm

6cm

A1

A2A3

A4

25cm2 20cm2

30 cm2x cm2

A

F E

DCB

G

8cm H

12cm2

8cm2

6cm2


Area and Volume (F.1)

Documents

total surface

measuring

finest markings

chapter 11amp

straight lines

hong kong

upper base

lower base