Chapter Module Solid Geometry III

8/6/2019 Chapter Module Solid Geometry III

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Prepared By :Prepared By :Pn Norisah bt MustaffaPn Norisah bt Mustaffa

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8.1 Volume Of Right Prisms and RightCircular Cilinders

8.2 Volume Of Right Pyramids AndCircular Cones

8.3 Volume Of Sphere

8.4 Volume Of Composite Solids

Prepared By:Norisah (KOSPINT)

Chapter 8Chapter 8


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8.1 (A) Volume Of Right Prisms

A prism is a solidwhich can be cutinto sections which

have the samecross-sectionalarea .

A cross-sectional A cross-sectional


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8.1 (A) i . Volume Of Right Prisms

The Formula to find the volume of a prism is,V = Area of cross-section x height (length)

length)

Cross-section (base)

A right prism is one in which the lateral edges are at right

angles to the bases.

Cross-section


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8.1 (A) ii . Calculating The Volume Of Right Prisms

V = Area of cross-section x height (length)

24 cm

Example 1: Find the volume of 24 cm long of prism having

a cross-sectional of a right angled triangle of base 8 cmand height 6 cm.

8 cm

6 cmV = ( 1 x 8 x 6 )x 24

2

= 576 cm 3

Solution:

Triangular Prism

Cross-section


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Example 2: Find the volume of the following solid that has

a quadrilateral cross-section and 6 cm height.

8.1 (A) iii . Calculating The Volume Of Right Prisms


V = [1

x (8+

5) x 6]x20

2

= 39 x 20

= 780

cm3

Solution:

Right Prism(trapezium)

20 cm

8 cm

5 cm

6 cm

Cross-section


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Example 3: The diagram below shows a right prism that

has a volume of 168 cm3

. Find the value of height (length).

8.1 (A) iv . Calculating The Height Of A Right Prism


(1 x 6 x 4 ) x l = 1 682

l = 1 68

12

l = 14 cm

Solution:5 cm

6 cm

5 cm4 cm

Length(l)

The height of a rightprism can be

calculated when thevolume and the

base area are given

Cross-section


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Example 4: Calculate the base area of a right prism thathas a volume of 896 cm 3 and height 16 cm.

8.1 (A) v . Calculating The Base Area(Cross-Section) Of A Right Prism

V = Area of cross-section x height (length)V = 89 6 cm 3

l (h) = 1 6 cm

Area(A) x l = V

A = Vl

A = 89 6 cm3

1 6 cm

= 56 cm 2

Solution:

The base area of a right prism can

be calculatedwhen the volume

and height aregiven .


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8.1 B (i) Volume Of Right Cylinder

A right circular cylinder is a solidwith a uniform andcircular cross-

section and its axisperpendicular toboth of ends .Circular

Cross-section

h

O radius


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Example 1: Find the volume of the following cylinder with

radius 7 cm and length of 21 cm.

8.1 B (ii) Culculating The Volume Of Right Cylinders


V = Tr 2 x height

= 22 x 7 x 7x 217

= 3 234 cm 3

Solution:

Cross-section

7cm21 cm

Area = Tr 2

V = Tr 2 h


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Tr 2 x height = V

22 x r 2 x 28 = 22007

r 2 = 2200 x 722 x 28

r 2 = 2 5

r = 2 5 = 5 cm

Example 2: Calculate the base radius of the cylinder with a

volume of 2 200 cm3

. and height 28 cm.

8.1 B (iii) Culculating The Radius Of A Right Cylinder


Solution:

28 cm

V = Tr 2 h

Cross-sectionArea = Tr 2

r


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8.1 C (i) Changing Units For VolumeVolume is a measurement of the space filled by a unit.Measurement in cubic unit .

1 cm 3 (cubic centimetres) = 1 000 mm 3 (cubic milimetres)1 m 3 (cubic metre) = 1 000 000 cm 3 (cubic centimetres)

E xample 1:Convert 0.84 m 3 in cm 3 .

= 0.84 x 1 000 000 cm 3

= 840 000 cm 3

E xample 2:Convert 243 000 mm 3 in cm 3 .

= 243 000 cm 3 1 000

= 243 cm 3

E xample 3:Convert 7 .9 cm 3 in mm 3 .

= 7 .9 x 1 000 mm 3

= 7 900 mm 3

E xample 2:Convert 6 1 5 200 cm 3 in m 3 .

= 6 1 5 200 cm 3 1 000 000

= 0. 61 52 m 3


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8.1 C (ii) Changing Units For Liquid VolumeVolume is a measurement of the space filled by a unit.Measurement unit for liquid volume .

1 l (liter) = 1 000 m l (milimetres)1 000 m l (mililiters) = 1 000 cm 3 (cubic centimetres)

E xample 1:Convert 1 56 cm 3 in m l 3 .

= 1 56 m l

E xample 2:Convert 14.8 l in m l .

= 14.8 x 1 000

= 14 800 m l

E xample 3:Convert 9 234 m l in l .

= 9 2341 000

= 9.234 l


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8.2 (A) Volume Of Right PyramidsA right pyramid is a solid with thesummit vertically above the midpoint of the base .The other faces are triangles thathave a common vertex called apex .

Volume of right pyramid,

= 1 x area of base x height3

= 1 x (l x w) x h3

w

apex

base

height (h)

l


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8.2 A (i) Calculating The Volume Of A Right Pyramid

Solution:

The diagram below shows a right pyramid that has a

rectangular base with length 18 cm and width 8 cm and itsheight is 12 cm. Find the volume.

18 cm

8 cm V = 1 x 18 x 8 x 123

= 576 cm 3The volume of a right pyramid

can be calculated when (a)height and area of the base, (b)

height and dimension of thebase of the pyramid are given.

V = 1 x Area of base x height3


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8.2 A (ii) Calculating The Height Of A Right Pyramid

Solution:

Given the volume of a right pyramid shown in the diagram

below is 252 cm3, what is the height.

V = 1 x Area of base x height3

1 x 12 x 8 x h = 2 52 cm 33

h = 2 52 x 312 x 8

h = 7 .88 cm

The height of a right pyramid can be calculated when the volume and dimension of the base of

the pyramid are given.

8 cm

12 cm


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1 x A x 1 5 = 7 20 cm3

3

A = 7 20 x 31 5

A = 144 cm 2

8.2 A (iii) Calculating The Area Of The BaseOf A Right Pyramid

Solution:

The volume of a right pyramid shown in the following

diagram is 720 cm 3, what is the area of its base?V = 1 x Area of base x height

31 5 cm

The area of the base of

a right pyramid can becalculated when thevolume and height of

the pyramid are given.


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8.2 (B) Volume Of Right Circular Cones

A right circular cone is a

solid that is the shape of the right pyramid but has acircular base and oneinclined surface .

Volume of right circular cones,= 1 x area of base x height

3

= 1 x Tr 2 x h3

= 1 Tr 2 h3

Circular base

Inclinedsurface

r

h


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8.2 B(i) Calculating The Volume Of A Right Cone

Calculate the volume of a cone with slant height 1 7 cmand height 1 5 cm .

Volume of right circular cones,= 1 x area of base x height

3

V = 1 Tr 2 h3

1 7 cm1 5 cm

8 cm

V = 1 Tr 2 h3

= 1 x 22 x 8 x 8 x 1 53 7

= 100 5 .7 cm 3

By using thePythagoras

Theorem, calculatethe value of radius

Solution:


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8.2 B(ii) Calculating The Height Of A Right Cone

If the volume of a cone is 1 232 cm 3 and the radius of its base is 7 cm . Calculate the height .

V = 1 Tr 2 h3

h cm

7 cm

1 Tr 2 h = 1 232 cm 33

h = 1 232 x 322 x 7 x 7

7h = 24 cm

The height of theright cone can becalculated whenthe volume and

radius of thebase are given .

Solution:


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8.2 B(iii) Calculating The Radius Of The Base Of A Right Cone

The volume of the right cone below is 3 69 6 cm 3. If itsheight is 14 cm, calculate the value of radius .

V = 1 Tr 2 h3

14 cm

r

1 Tr 2 h = 3 69 6 cm 33

r 2

= 3 69 6 x 322 x 147

r 2 = 2 52

r = 2 52 = 1 5 .8 7 cm

The radius of theright cone can be

calculated when thevolume and height

of the cone aregiven .

Solution:


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8.3 Volume Of A Sphere

A sphere is a solid of

which the points on itssurface areequidistance from itscentre .

Volume of sphere= 4 x Tr 3

3

= 4 Tr 33

r

Volume of hemisphere

= 1 x ( 4 T j3 )2 3

= 2 T j33

r r


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8.3 A (i) Culculating The Volume Of A Sphere

A solid sphere made of iron has a radius of 21 cm . Find its volume .

Volume of sphere

= 4 Tr 3

3

= 4 x 22 x 21 x 21 x 213 7

= 38 808 cm 3

Solution:

21 cm


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8.3 A (ii) Culculating The Volume Of A Sphere

Calculate the radius of a sphere that has a volume of 143 7 1 cm 3 .

3

Radius of sphere,4 Tr 3 = 143 7 1 cm 33 3 4 x 22 x r 3 = 43123 7 3

r 3 = 4312 x 3 x 73 x 4 x 22

= 343r = 3 343

= 7 cm

Solution:

r


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8.3 A (iii) Culculating The Volume Of A Hemisphere

Calculate the volume of a hemisphere that has adiameter of 14 cm .

Volume of hemisphere,V = 2 Tr 3

3 = 2 x 22 x 7 x 7 x 73 7

= 7 18 2 cm 33

Solution:

r r


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8.4 Composite SolidsA composite solid is a combination

of geometric solids .

Prism

CubePyramid

Coboid

Cylinder

Cylinder

ConeHemisphere


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8.4 Culculating The Volume Of Composite SolidsThe diagram below shows a solid that is composedof a cuboid and a pyramid . Calculate the volume of the solid .

V = ( l x w x h) + 1 (l x w) x h

3 = ( 8 x 2 x 4 ) + (1 x 8 x 2 x 12 )

3 = 6 4 + 64= 128 cm 3

Solution:

4 cm

2 cm8 cm

1 5 cm

9 cm

h = 12 cm(Pythagoras

Theorem)

12 cm

Area of thebase , length x

width

Height or length


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8.4 Culculating The Volume Of A Composite SolidsThe diagram below shows a solid that is composedof a hemisphere and a cone . Calculate the volume of the solid .

V = 2 Tr 3 + 1 Tr 2 h3 3

=(2 x22 x7x7x7) + (1 x22 x7x7x7)3 7 3 7

= 21 56 + 10 783 3

= 32343

= 10 78 cm 3

Solution:

14 cm

14 cm


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8.4 Culculating The Volume Of A Composite SolidsThe diagram below shows a solid that is composedof a cylinder and two hemisphere . Calculate thevolume of the solid .

V = 4 Tr 3 + Tr 2 h3

=(4 x22 x6x6x6) + (22 x6x6x 14 )3 7 7

= 6 33 6 + 1 584

7= 2 489 1 cm 3

7

Solution:

14 cm

r

r

6 cm

2 6 cm


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8.4 Culculating The Volume Of A Composite SolidsThe diagram below shows a solid that is composedof prism and a cuboid . Calculate the volume of thesolid .

V = 1 (l x w) h + (l x w x h)3

= ( 1 x1 6x 12 x6) + (1 6 x 6 x 10 )3

= 384 + 9 60

= 1 344 cm3

Solution:

6 cm

6 cm

10 cm

1 6 cm

1 cm

Area of thebase = length

x width

Height or length


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If mathematically you end up with the wrong answer, try

multiplying by the page number .

Chapter Module Solid Geometry III

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