Area and Volume [I]

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Area and Volume I
9
Case Study
9.1 Areas of Polygons
9.2 Volumes and Total Surface Areas of Prisms
Chapter Summary
2
Case Study
? Perimeter of rectangle ? (Length ?
Width) ? 2
(Length ? Width) ? 2 ? 20
? Length ? Width ? 10
Length ? Width
? If the wire is bent into a square of side
length 5 cm, it will attain the largest area of
25 cm2.
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9.1 Areas of Polygons
In primary level, we learnt how to find the area
of different polygons by using the
area-dissecting algorithm and the area-filling
algorithm.
Area of the figure ? Area of square I ? Area
of triangle II
Area of the figure ? Area of rectangle ? Area
of trapezium IV
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Example 9.1T
9.1 Areas of Polygons
In the figure, find the area of the polygon
ABCDEF.
Solution (area-dissecting algorithm)
From E, construct a line EG such that EG ? BC.

• Area of the polygon
• ? Area of rectangle ABGF ? Area of rectangle EGCD

? 6 ? 3 ? 4 ? (10 ? 3) cm2
? (18 ? 28) cm2
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Example 9.1T
9.1 Areas of Polygons
In the figure, find the area of the polygon
ABCDEF.
Solution (area-filling algorithm)
Extend the lines AF and CD to meet at K such that
FK ? DK.

• Area of the polygon
• ? Area of rectangle ABCK ? Area of rectangle FEDK

? 6 ? 10 ? (6 ? 4) ? (10 ? 3) cm2
? (60 14) cm2
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Example 9.2T
9.1 Areas of Polygons
In the figure, AB ? 6 cm, BC ? 8 cm and AC ? 10
cm. Find the value of the unknown a.
D
Solution
The area calculated by the two methods must be
the same, i.e., 5a ? 24
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Example 9.3T
9.1 Areas of Polygons
In the figure, find the area of the
polygon ABCDEGH.
Solution
Area of the polygon ? Area of rectangle ABCH ?
Area of parallelogram CDGH ? Area of triangle
DEG
In this example, it is better for us to use the
area-dissecting algorithm.
If we use the area-filling algorithm instead, the
calculation will become more complicated and some
information may not be given as well.
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Example 9.4T
9.1 Areas of Polygons

• The figure shows a metal plate.
• (a) Find the area of the metal plate.
• The cost of polishing the metal plate is 0.2/cm2.
  
• Find the total cost of polishing the
  metal plate.

Solution
(a) Area of the metal plate ? Area
of square ? Area of triangle X ? Area of triangle
Y
(b) Total cost ? (0.2 ? 112.5)
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9.2 Volumes and Total Surface Areas of
Prisms
Prisms are 3-D solids with uniform cross-sections
which are in form of a polygon.
For a prism,
? bases 2 end faces ? height distance
between the 2 bases ? lateral faces faces
adjacent to the bases
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9.2 Volumes and Total Surface Areas of
Prisms
We name a prism according to the shape of its
base. For example
Cubes and cuboids are kinds of rectangular
prisms.
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9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
In primary level, we learnt the formulas for
finding the volumes of cubes and cuboids.

• Volume of cube
• ? Length ? Length ? Length
• ? Base area ? Height

• Volume of cuboid
• ? Length ? Width ? Height
• ? Base area ? Height

The volume of a prism equals to the product of
its base area and its height.
Volume of a prism ? Base area ? Height
Note The unit of volume is cubic unit, that is
mm3, cm3, m3, etc.
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Example 9.5T
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
In the figure, find the volume of the prism.
Solution
The base of the prism is formed by a square and a
trapezium.
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Example 9.6T
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
In the figure, the volume of the block is 44 cm3.
Find the value of x.
Solution
The base of the block is formed by a square and a
trapezium.
Volume of the block ? Base area ? Height
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Example 9.7T
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
The figure shows a cup. Daniel fills the cup with
water such that 1 cm of height is left
unfilled. (a) Find the volume of the
water. (b) Daniel adds 4 metal balls of volume 4
cm3 each into the water. Find the rise in
the water level.
Solution

1. The base of the cup is formed by a rectangle and
   a trapezium.

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Example 9.7T
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
The figure shows a cup. Daniel fills the cup with
water such that 1 cm of height is left
unfilled. (a) Find the volume of the
water. (b) Daniel adds 4 metal balls of volume 4
cm3 each into the water. Find the rise in
the water level.
Solution

1. Let h cm be the rise in the water level.

Volume of metal balls ? Volume of water rise
? The rise in the water level is 0.25 cm.
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Example 9.8T
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
A piece of metal as shown in the figure is melted
and recast into another rectangular metal. (a)
What is the volume of the metal? (b) Find the
value of a.
Solution
(b) Volume of the rectangular metal ? (5 ? 6 ?
a) cm3
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9.2 Volumes and Total Surface Areas of
Prisms
B. Total Surface Areas of Prisms
The total surface area of a prism is the sum of
the areas of all its faces.
That is, the total surface area of a prism equals
the total area of the 2 bases and the lateral
faces.
Total surface area of a prism ? 2 ? Base area
? Total area of the lateral faces
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Example 9.9T
9.2 Volumes and Total Surface Areas of
Prisms
B. Total Surface Areas of Prisms
Find the total surface area of the prism.
Solution
Area of the base of the prism ? (5 ? 3) cm2
? 15 cm2
Total area of the lateral faces ? (3.5 ? 8 ? 2 ?
5 ? 8 ? 2) cm2
? 136 cm2
Total surface area of the prism ? (15 ? 2 ? 136)
cm2
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Example 9.10T
9.2 Volumes and Total Surface Areas of
Prisms
B. Total Surface Areas of Prisms
The figure shows an open box that was made from
paper. Find the total area of the paper used to
make it.
Solution
? 36 cm2
Total area of 3 lateral faces ? (5 ? 12 ? 6 ? 12
? 5 ? 12) cm2
? 192 cm2
Total area of paper used ? (36 ? 2 ? 192) cm2
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Chapter Summary
9.1 Areas of Polygons
Areas of polygons can be calculated by the
following 2 methods.
1. Area-dissecting algorithm
2. Area-filling algorithm
9.2 Volumes and Total Surface Areas of Prisms
1. Volume of a prism ? Base area ? Height
2. Total surface area of a prism ? 2
? Base area ? Total area of the lateral faces
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Follow-up 9.1
9.1 Areas of Polygons
In the figure, find the area of the polygon
ABCDE.
Solution (area-dissecting algorithm)
From E, construct a line EF such that EF ? BC.
Area of the polygon ? Area of rectangle ABFE ?
Area of trapezium CDEF
? (12 ? 12) cm2
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9.1 Areas of Polygons
Follow-up 9.1
In the figure, find the area of the polygon
ABCDE.
Solution (area-filling algorithm)
Extend the lines AE and CD to meet at K such that
EK ? DK.

• Area of the polygon
• ? Area of rectangle ABCK ? Area of triangle EDK

? (28 ? 4) cm2
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Follow-up 9.2
9.1 Areas of Polygons
In the figure, QR ? 16 cm, SR ? 10 cm and SU ? 8
cm. Find the value of the unknown h.
Solution
Area of parallelogram PQRS with base RS ? RS ?
US ? (10 ? 8) cm2 ? 80 cm2
Area of parallelogram PQRS with base QR ? QR ?
VR ? (16 ? h) cm2 ? 16h cm2
The area calculated by the two methods must be
the same, i.e., 16h ? 80
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Follow-up 9.3
9.1 Areas of Polygons
In the figure, find the area of the
polygon ABCDEFG.
Solution
Area of the polygon ? Area of trapezium CDEF ?
Area of square BCFG ? Area of triangle ABG
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Follow-up 9.4
9.1 Areas of Polygons

• The figure shows a garden with 2 fish ponds A and
  B.
• (a) Find the planted area of the garden.
• If the cost of planting flowers is 40/m2, find
  the
• total cost of planting flowers in the
  garden.

Solution
(a) Area of the garden ? Total
area of the garden and the 2 fish ponds
? Area of triangle A ? Area of trapezium B
(b) Total cost ? (40 ? 16)
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Follow-up 9.5
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
In the figure, find the volume of the prism.
Solution
The base of the prism is formed by a triangle and
a rectangle.

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Follow-up 9.6
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
In the figure, the volume of the brick in the
shape of a prism is 1700 cm3. Find the value of
d.
Solution
The base of the brick is formed by a square and a
trapezium.
Volume of the brick ? Base area ? Height

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Follow-up 9.7
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms

• The figure shows a bottle of ink.
• (a) Find the volume of the ink in the bottle.
• Kenneth adds 8 drops of ink into the bottle.
• Suppose that the volume of each drop of
  ink
• is 0.2 cm3. Find the rise in the level of
  ink.

Solution

1. The base of the bottle is formed by a
   rectangle and a trapezium.

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Follow-up 9.7
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms

• The figure shows a bottle of ink.
• (a) Find the volume of the ink in the bottle.
• Kenneth adds 8 drops of ink into the bottle.
• Suppose that the volume of each drop of
  ink
• is 0.2 cm3. Find the rise in the level of
  ink.

Solution
(b) Let b cm be the rise in the level of ink.
Volume of 8 drops of ink ? Volume of ink rise

? The rise in the level of ink is 0.4 cm.
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Follow-up 9.8
9.2 Volumes and Total Surface Areas of
Prisms
A. Volume of Prisms
The figure shows a chocolate with a parallelogram
base. Kathy melts the chocolate and moulds it
into cubes of side length 2 cm. (a) What is
the volume of the chocolate? (b) How many
chocolate cubes can she make?
Solution
(b) Let c be the number of chocolate tubes she
can make.
? She can make 12 chocolate cubes.
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Follow-up 9.9
9.2 Volumes and Total Surface Areas of
Prisms
B. Total Surface Areas of Prisms
Find the total surface area of the prism.
Solution
? 26 cm2
Total area of the lateral faces ? Area of ADHE
? Area of AEFB ? Area of BCGF ? Area of DCGH

? (5 ? 7 ? 5 ? 7 ? 8 ? 7 ? 5 ? 7) cm2
? 161 cm2
Total surface area of the prism ? (26 ? 2 ? 161)
cm2
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Follow-up 9.10
9.2 Volumes and Total Surface Areas of
Prisms
B. Total Surface Areas of Prisms
Calvin made a folder as shown in the figure.Find
the total area of the paper used to make the
folder.
Solution
? 54 cm2
Total area of the lateral faces ? (5 ? 8 ? 6 ? 8
? 13 ? 8) cm2
? 192 cm2
Total area of paper used ? (52 ? 2 ? 192) cm2