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1
Topic 4
Volumes of Solids
Contents
4
...
4
...
2
7
4
...
4
...
12
15
4
...
17
Learning Objectives
By the end of this topic, you should be able to:
• calculate the volume of a sphere, cone, pyramid, prism;
• calculate an unknown dimension given the volume of a solid;
• calculate the volume of composite solids
...
VOLUMES OF SOLIDS
4
...
Here is a summary of the formulae for the volume of the solids we have met
...
VOLUMES OF SOLIDS
3
Volume of a Cuboid =
lbh
Volume of a Cylinder =
πr 2 h
Volume of a Cone =
Volume of a pyramid =
1 2
πr h
3
1
× Area of the base × height
3
Volume of a Sphere =
4 3
πr
3
Volume of a prism =
area of the cross − sec tion × length
Examples
1
...
Calculate its volume correct to 1 decimal
place
...
Substitute
...
Round
...
p
...
2
...
Calculate its volume correct to 1 decimal
place
...
V = 1 /3 πr2 h
Substitute
...
Round
...
p
...
VOLUMES OF SOLIDS
...
Problem:
Calculate the volume of the square
based pyramid
...
V = 1 /3 × area of base ×
height
Identify the base
...
V = 1 /3 × 142 × 12
Evaluate
...
4
...
Calculate its volume correct to 1 decimal
place
...
Substitute
...
Round
...
p
...
5
...
Solution:
© H ERIOT-WATT U NIVERSITY
TOPIC 4
...
V = 1 /2 × area of cross-section × length
Identify the cross-section
...
V = 1 /3 × 12 × 9 × 25
Evaluate
...
Calculating the volume of standard solids practice
Q1:
A cylinder has a radius of 8 cm and a
height of 5 cm
...
...
Calculate its volume correct to 1 decimal
place
...
Q3:
A rectangular pyramid has a length of 16
m, a breadth of 18 m and a height of
23
...
Calculate its volume correct to 1 decimal
place
...
Q4:
A sphere has a radius of 8 cm
...
...
VOLUMES OF SOLIDS
Q5:
A triangular prism has a length of 1 m, a
triangle base of 36 cm and a height of
22·5 cm
...
...
Find the volume of a cylinder with a
radius of 5 m and a height of 7 m
...
...
Find the volume of a cone with a radius
of 6cm and a height of 3 cm
...
Q8:
The formula for the volume of a sphere is
V = 4 /3 πr3
where r is the radius of the sphere
...
What is the volume of the sphere, in cm 3 ,
correct to the nearest whole number?
...
VOLUMES OF SOLIDS
Q9:
The tent is in the shape of a triangular
prism
...
The tent has a height of 0
...
Calculate the volume of triangular prism,
correct to 3 significant figures
...
Q10:
The picture shows the Great Pyramid of
Giza in Egypt
...
It is a square-based pyramid of length
230·4 m and height 146·4 m
...
...
2
Finding an unknown dimension when you know the
volume
By re-arranging the formula for the volume we can find an unknown dimension
...
Problem:
If the volume of the cylinder is 4063 cm3 ,
calculate it's height
...
Solution:
The formula for the volume of a cylinder is V = πr 2 h where radius r = 8·5 cm
...
VOLUMES OF SOLIDS
4063 /
226·98
=h
h = 17·9 cm (to 1d
...
)
...
Problem:
If the volume of the sphere is 14·1 m 3 ,
calculate it's radius
...
Solution:
The formula for the volume of a sphere is V = 4 /3 πr3
...
p
...
3
...
Give your answer to 1 decimal place
...
If we substitute the volume and the height we get
© H ERIOT-WATT U NIVERSITY
TOPIC 4
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p
...
4
...
Solution:
The formula for the volume of a triangular prism is V = area of the cross section × length
= 1 /2 × l × b × h with height h = 42 mm and base b = 34 mm
...
5
...
Solution:
The formula for the volume of a triangular pyramid is V = 1 /3 × area of base × height
The base of the pyramid is a triangle and the formula for the area of a triangle is
A= 1 /2 bh
...
VOLUMES OF SOLIDS
If we substitute the volume, base and height of the triangle we get
14560 =
1 1
× × 40 × 42 × h
3 2
14560 = 280 × h
14560
=h
280
h = 52 cm
1 1
× × 40 × 42 = 280
3 2
(re − arrange the equation)
...
Give your answer to 1 decimal place
...
Q12:
Calculate the radius of the sphere if it's
volume is 7·2 m3
...
...
Give your answer to 1 decimal place
...
© H ERIOT-WATT U NIVERSITY
TOPIC 4
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Give your answer to 1 decimal place
...
Q15:
Calculate the length of the base of the
square-based pyramid if it's volume is
5203 m3
...
...
The volume of the tin of beans is 507·5
cm3
...
7 cm, calculate the radius of the tin
giving your answer correct to 1 decimal
place
...
© H ERIOT-WATT U NIVERSITY
11
12
TOPIC 4
...
The volume of ice-cream which this cone
could contain is 81·8 ml
...
Give your answer correct to 3 significant
figures
...
Q18: Finding the radius of a sphere
...
The volume of
our planet is 1·08321 × 10 12 km3
...
...
The pyramid has a rectangular base of
length 102 m and breadth 84 m
...
Give your answer correct to 3 significant
figures
...
4
...
Examples
1
...
VOLUMES OF SOLIDS
The pencil is 21 cm long and has a
radius of 0·5 cm
...
Solution:
This shape is made from a cylinder and a cone and both shapes have the same radius
...
2
...
Calculate the volume of the obelisk,
correct to 3 significant figures
...
We need Volume of a Cuboid = lbh and Volume of a pyramid = 1 /3 ×
Area of the base × height
The height of the pyramid = 10·8 - 9 = 1·8 m
Volume of a Cuboid = 2·6 × 2·6 × 9 = 60·84 cm 3
Volume of a pyramid = 1 /3 × 2·62 × 1·8 = 4·056 cm3
Total Volume = 60·84 + 4·056 = 64·896
= 64·9 cm3 to 3 significant figures
...
Problem:
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13
14
TOPIC 4
...
Calculate the volume of space in the box
...
The side of the cube is the same as
the diameter of the sphere
...
Volumes of composite solids exercise
Q20:
The salt shaker is 47mm tall and has a
radius of 16 mm
...
(Hint: the
radius of the cylinder and hemisphere
are the same
...
Q21:
A triangular prism of chocolate is placed
in a cuboid
...
Calculate the volume of space left in the
box correct to 3 significant figures
...
© H ERIOT-WATT U NIVERSITY
TOPIC 4
...
The cone has a radius of 5 cm and a
height of 16 cm
...
...
Calculate the height of the pyramid,
correct to 1 decimal place
...
Q24:
A 10 cm cube of chocolate is melted
down
...
2 cm can be formed from the
chocolate?
(Hint: Do not round your answer for the
volume of a sphere of chocolate
...
4
...
VOLUMES OF SOLIDS
• Volume of a pyramid =
1
× Area of the base × height
3
• Volume of a Sphere =
4 3
πr
3
• Volume of a prism =
area of the cross − sec tion × length
• You may need the Area of a Triangle =
1
bh
2
Area of a Square =
l2
and Area of a Rectangle =
lb
• To find an unknown dimension on a solid shape when you know it's volume:
– put the values you know into the formula including the volume
– rearrange or change the subject of the formula
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TOPIC 4
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5
End of topic test
End of topic 4 test
Q25:
Volume of a Cylinder
The formula for the volume of a cylinder
is
V = πr2 h
where r is the radius and h is the height
of the cylinder
...
What is the volume of the cylinder in m 3 ?
Give your answer correct to 1 decimal
place
...
Q26:
Volume of a Cone
The formula for the volume of a cone is
V = 1 /3 πr2 h
where r is the radius and h is the height
of the cone
...
What is the volume of the cone, in cm 3 ,
correct to 2 decimal places?
...
A sphere has a radius of 7 cm
...
© H ERIOT-WATT U NIVERSITY
17
18
TOPIC 4
...
Calculate the volume of the paperweight
...
...
It is 52 cm tall and has a triangular base
of side 30 cm and height 26 cm
...
...
If the
radius is 5·2 cm, calculate the height of
the beaker
...
(Hint: 1 ml = 1 cm3 )
...
...
VOLUMES OF SOLIDS
Q32:
Volume of a Composite Solid
A cone and a sphere are placed inside a
cylinder
...
a) Calculate the volume of the cylinder
correct to 5 significant figures
...
c) Calculate the volume of the sphere
correct to 4 significant figures
...
...
cm 3
Number of whole spheres = 1105
End of topic 4 test (page 17)
Q25: 791·7 m 3
Q26: 209·44 cm 3
Q27: 1437 cm3
Q28: 23500 mm 3
Q29: 20280 cm 3
Q30: 16·5 cm3
Q31: 25 cm
Q32:
a) 9802 cm3
b) 2112 cm3
c) 2310 cm3
d) 5380 cm3
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