Mathematics SKE, Strand C
UNIT C4 Volumes: Text
STRAND C: Measurement
C4 Volumes
Text
Contents
Section
C4.1
* C4.2
* C4.3
* C4.4
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Volumes of Cubes, Cuboids, Cylinders and Prisms
Mass, Volume and Density
Volumes of Pyramids, Cones and Spheres
Dimensions
Mathematics SKE, Strand C
UNIT C4 Volumes: Text
C4 Volumes
C4.1 Volumes of Cubes, Cuboids, Cylinders
and Prisms
The volume of a cube is given by
a
V = a3
a
a
where a is the length of each side of the cube.
For a cuboid the volume is given by
V = abc
b
c
where a, b and c are the lengths shown in the diagram.
a
h
The volume of a cylinder is given by
V = π r2 h
r
where r is the radius of the cylinder and h is its height.
The volume of a triangular prism can be expressed in two ways,
as
V = Al
where A is the area of the end and l the length of the prism,
or as
1
V = bhl
2
where b is the base of the triangle and h is the height of the triangle.
Worked Example 1
The diagram shows a truck.
4m
2.5 m
2m
Find the volume of the load-carrying part of the truck.
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1
l
h
A
b
C4.1
UNIT C4 Volumes: Text
Mathematics SKE, Strand C
Solution
The load-carrying part of the truck is represented by a cuboid, so its volume is given by
V = 2 × 2.5 × 4
= 20 m 3
Worked Example 2
The cylindrical body of a fire extinguisher has the
dimensions shown in the diagram. Find the maximum
volume of water the extinguisher could hold.
Solution
60 cm
The body of the extinguisher is a cylinder with
radius 10 cm and height 60 cm, so its volume is
given by
V = π × 10 2 × 60
= 18850 cm 3
(to the nearest cm3)
20 cm
Worked Example 3
A traffic calming road hump (sleeping policeman) is made of concrete and has the
dimensions shown in the diagram. Find the volume of concrete needed to make one road
hump.
10 cm
300 cm
80 cm
Solution
The shape is a triangular prism with b = 80, h = 10 and l = 300 cm. So its volume is
given by
1
V = × 80 × 10 × 300
2
= 120 000 cm3
Worked Example 4
The diagram below, not drawn to scale, shows a container in the shape of a rectangular
prism.
h
40 cm
75 cm
The base of the container has a length of 75 cm and a width of 40 cm.
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Mathematics SKE, Strand C
C4.1
(i)
UNIT C4 Volumes: Text
Calculate the area, in cm 2 , of the base of the container.
Water is poured into the container, reaching a height of 15 cm.
(ii)
(iii)
Calculate, in cm 3 , the volume of water in the container.
If the container holds 84 litres when full, calculate the height, h, in cm, of the
water when the container is full.
Solution
(i)
Area of base = 75 × 40 = 3000 cm 2
(ii)
Volume
= 15 × 3000 cm 2
= 45 000 cm 3
(iii)
When full, the tank holds 84 × 1000 cm 3 of water, so
h × 3000
= 84 000
h =
84 000
3000
= 28 cm
Exercises
1.
Find the volume of each solid shown below.
(a)
(b)
5m
10 cm
5 cm
12 cm
3 cm
5 cm
(c)
(d)
8 mm
1.4 m
2m
20 mm
3.2 m
0.5 m
30 cm
(e)
(f)
8 cm
2.5 m
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C4.1
Mathematics SKE, Strand C
UNIT C4 Volumes: Text
(g)
(h)
8 cm
12 cm
12 cm
2
Area = 42 cm
3 cm
(i)
1m
3.2 m
1.5 m
2.
(a)
(b)
Find the volume of the litter bin shown in
the diagram, in m3 to 2 decimal places.
90 cm
Find the volume of rubbish that can be put in
the bin, if it must all be below the level of the
hole in the side, in m3 to 2 decimal places.
70 cm
60 cm
3.
A water tank has the dimensions shown in
the diagram.
(a)
Find the volume of the tank.
(b)
If the depth of water is 1.2 m, find the
volume of the water.
1.4 m
2.3 m
1.2 m
4.
A concrete pillar is a cylinder with a radius of 20 cm and a height of 2 m.
(a)
Find the volume of the pillar.
The pillar is made of concrete, but contains 10 steel rods of length 1.8 m and
diameter 1.2 cm.
5.
(b)
Find the volume of one of the rods and the volume of steel in the pillar.
(c)
Find the volume of concrete contained in the pillar.
The box shown in the diagram contains
chocolate.
(a)
Find the volume of the box.
(b)
If the box contains 15 cm3 of air,
find the volume of the chocolate.
4 cm
20 cm
3 cm
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Mathematics SKE, Strand C
C4.1
6.
UNIT C4 Volumes: Text
Find the volume of each prism below.
(a)
(b)
3 cm
1 cm
1 cm
1 cm
1 cm
1 cm
2 cm
1 cm
1 cm
24 cm
3 cm
15 cm
2 cm
(c)
(d)
1 cm
2 cm
3 cm
8 cm
4 cm
40 cm
4 cm
6 cm
22 cm
1 cm
7.
Each diagram below shows the cross section of a prism. Find the volume of the
prism, given the length specified.
(a)
(b)
Length 20 cm
Length 40 cm
1 cm
3 cm
7 cm
5 cm
2 cm
5 cm
2 cm
8.
The diagram shows the cross section of a length of guttering. Find the maximum
volume of water that a 5 m length of guttering could hold.
10 cm
5 cm
6 cm
2m
9.
The diagram shows the cross section of a
skip that is 15 m in length and is used to
deliver sand to building sites. Find the
volume of sand in the skip when it is
filled level to the top.
0.5 m
2.5 m
1.5 m
2m
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UNIT C4 Volumes: Text
Mathematics SKE, Strand C
C4.1
10.
A ramp is constructed from concrete. Find the volume of concrete contained in the
ramp.
Not to scale
1.5 m
4m
5m
2.1 m
11.
2.3 m
O
RG ER
A
C AIN
NT
CO 5.4 m
Not to scale
The diagram shows a cargo container.
Calculate the volume of the container.
12.
30 cm
E
D
9 cm
F
7 cm
A
C
16 cm
B
The diagram above, not drawn to scale, shows ABCDEF, a vertical cross-section
of a container with ED being the top edge. DC and EF are vertical edges.
BC and AF are arcs of a circle of radius 7 cm and AB
E D.
E D = 30 cm ; AB = 16 cm ; E F = DC = 9 cm .
(i)
(ii)
13.
22
, show that the area of ABCDEF is 459 cm 2 .
7
Water is poured into the container until the water level is 4 cm from the top.
If the container is 40 cm long and has uniform cross-section, calculate, to the
nearest litre, the volume of water in the container.
Taking π =
Tomato soup is sold in cylindrical tins.
12 cm
Each tin has a base radius of 3.5 cm and a height of 12 cm.
TOMATO
SOUP
3.5 cm
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Not to scale
C4.1
UNIT C4 Volumes: Text
Mathematics SKE, Strand C
(a)
Calculate the volume of soup in a full tin.
Take π to be 3.14 or use the π key on your calculator.
(b)
Bradley has a full tin of tomato soup for dinner. He pours the soup into a
cylindrical bowl of radius 7 cm.
Not to scale
7 cm
What is the depth of the soup in the bowl?
14.
1m
25 m
16 m
2.7 m
Diagram NOT
accurately drawn
8m
5m
The diagram represents a swimming pool.
The pool has vertical sides.
The pool is 8 m wide.
(a)
Calculate the area of the shaded cross section.
The swimming pool is completely filled with water.
(b)
Calculate the volume of water in the pool.
64 m3 leaks out of the pool.
(c)
15.
Calculate the distance by which the water level falls.
The diagram represents a carton in the shape of a cuboid.
12.5 cm
5 cm
8 cm
(a)
Calculate the volume of the carton.
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UNIT C4 Volumes: Text
Mathematics SKE, Strand C
C4.1
There are 125 grams of sweets in a full carton.
John has to design a new carton that will contain 100 grams of sweets when it is full.
(b)
(i)
Work out the volume of the new carton.
(ii)
Express the weight of the new carton as a percentage of the weight of
the carton shown.
The new carton is in the shape of a cuboid.
The base of the new carton measures 7 cm by 6 cm.
(c)
(i)
Work out the area of the base of the new carton.
(ii)
Calculate the height of the new carton.
C4.2 Mass, Volume and Density
Density is a term used to describe the mass of a unit of volume of a substance. For
example, if the density of a metal is 2000 kg/m3, then 1 m3 of the substance has a mass of
2000 kg.
Mass, volume and density are related by the following equations.
Mass
= Volume × Density
Volume =
Mass
Density
Density =
Mass
Volume
and
The density of water is 1 gram / cm3 (or 1 g cm −3 ) or 1000 kg / m3 (or 1000 kg m −3 ).
Worked Example 1
Find the mass of water in the fish tank shown in the diagram.
20 cm
30 cm
25 cm
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C4.2
Mathematics SKE, Strand C
UNIT C4 Volumes: Text
Solution
First calculate the volume of water.
V = 25 × 30 × 20
= 15000 cm 3
Mass
Now use
= Volume × Density
= 15000 × 1
(as density of water = 1g / cm3)
= 15000 grams
= 15 kg
Worked Example 2
The block of metal shown has a mass of 500 grams.
Find its density in
(a)
g / cm3,
(b)
kg / m3.
5 cm
Solution
(a)
10 cm
8 cm
First find the volume.
Volume = 5 × 8 × 10
= 400 cm 3
Then use
Density =
Mass
Volume
Density =
500
400
= 1.25 g /cm3
(b)
The process can then be repeated working in kg and m.
Volume = 0.05 × 0.08 × 0.1
= 0.0004 m3
Density =
=
0.5
0.0004
5000
4
= 1250 kg / m3
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Mathematics SKE, Strand C
C4.2
UNIT C4 Volumes: Text
Exercises
1.
2.
A drinks carton is a cuboid with size as shown.
(a)
Find the volume of the carton.
(b)
If it contains 8 cm3 of air, find the volume
of the drink.
(c)
Find the mass of the drink if it has a density
of 1 gram / cm3.
8 cm
6 cm
2 cm
20 cm
The diagram shows a concrete block of mass 6 kg.
(a)
Find the volume of the block.
(b)
What is the density of the concrete?
40 cm
10 cm
3.
21 cm
A ream (500 sheets) of paper is shown in the
diagram.
5 cm
If the mass of the ream is 2.5 kg, find the
density of the paper.
4.
5.
30 cm
A barrel is a cylinder with radius 40 cm and height 80 cm. It is full of water.
(a)
Find the volume of the barrel.
(b)
Find the mass of the water in the barrel.
A metal bar has a cross section with an area of 3 cm2 and a length of 40 cm. Its
mass is 300 grams.
(a)
Find the volume of the bar.
(b)
Find the density of the bar.
(c)
Find the mass of another bar with the same cross section and length 50 cm.
(d)
Find the mass of a bar made from the same material, but with a cross section
of area 5 cm2 and length 80 cm.
6.
A bottle which holds 450 cm3 of water has a mass of 530 grams. What is the mass
of the empty bottle?
7.
The diagram shows the dimensions of a swimming pool.
(a)
Find the volume of the swimming pool.
(b)
Find the mass of water in the pool if it
is completely full.
(c)
In practice, the level of the water is
20 cm below the top of the pool.
5m
2m
Find the volume and mass of the water in this case.
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8m
1m
Mathematics SKE, Strand C
C4.2
8.
9.
10.
UNIT C4 Volumes: Text
The density of a metal is 3 grams / cm3. It is used to make a pipe with external
radius of 1.5 cm and an internal radius of 0.5 cm.
(a)
Find the area of the cross section of the pipe.
(b)
If the length of the pipe is 75 cm, find its mass.
(c)
Find the length, to the nearest cm, of a pipe
that has a mass of 750 grams.
A foam ball has a mass of 200 grams and a radius of 10 cm.
4π r 3
to find the volume of the ball.
3
(a)
Use the formula V =
(b)
Find the density of the ball.
(c)
The same type of foam is used to make a cube with sides of length 12 cm.
Find the mass of the cube.
The diagram shows the cross section of a metal beam. A 2 m length of the beam has
a mass of 48 kg.
(a)
Find the density of the metal.
10 cm
A second type of beam uses the same type of metal,
but all the dimensions of the cross section are
increased by 50%.
(b)
1 cm
2 cm
12 cm
1 cm
Find the length of a beam of this type that
has the same mass as the first beam.
C4.3 Volumes of Pyramids, Cones and
Spheres
The volumes of a pyramid, a cone and a sphere are found using the following formulae.
Pyramid
Cone
Sphere
r
h
A
h
r
1
1
4
Ah
V = π r 2h
V = π r3
3
3
3
The proofs of these results are rather more complex and require mathematical analysis
beyond the scope of this text.
V=
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Mathematics SKE, Strand C
C4.3
UNIT C4 Volumes: Text
Worked Example 1
A cone and sphere have the same radius of 12 cm. Find the height of the cone if the cone
and sphere have the same volume.
Solution
Suppose that the height of the cone is h cm.
Volume of cone
1
= π × 12 2 × h = 48π h cm 3
3
Volume of sphere =
4 3
4
π r = π (12)3 = 2304π cm 3
3
3
Since the volumes are equal
48π h = 2304π
Solving for h,
h=
2304 π
2304
=
= 48 cm
48 π
48
Exercises
1.
Find the volume of the following containers:
(a)
a cylinder of base radius 3 cm and height 10 cm,
(b)
a cone of base radius 5 cm and height 12 cm,
(c)
a cone of base radius 7 cm and height 5 cm,
(d)
a cone of base radius 1 m and height 1.5 m,
(e)
a cone of base radius 9 cm and slant height 15 cm,
(f)
a cone of base diameter 20 cm and slant height 25 cm,
(g)
a sphere of radius 6 cm,
(h)
a hemisphere of radius 5 cm,
(i)
a pyramid of square base 10 cm and height 15 cm,
(j)
a pyramid of rectangular base 8 cm by 6 cm and height 9 cm.
2.
Calculate the radius of a sphere which has the same volume as a solid cylinder of
base radius 5 cm and height 12 cm.
3.
A wine glass is in the shape of a cone
on a stem. The cylindrical tumbler is
used to fill the wine glass. Find the
diameter of the tumbler so that it has
the same volume as the wine glass.
6 cm
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10 cm
12 cm
C4.3
Mathematics SKE, Strand C
4.
UNIT C4 Volumes: Text
Find the volume of the following shapes made
up of cones, hemispheres and cylinders.
(a)
(b)
6 cm
7 cm
6 cm
11 cm
(c)
4 cm
(d)
5.2 cm
7.1 cm
3.1 cm
11.2 cm
5.1 cm
5.
A cylinder is half filled with water as shown.
A heavy sphere of diameter 2.5 cm is placed
in the cylinder and sinks to the bottom.
By how much does the water rise in the cylinder?
7 cm
2.4 cm
6.
A test tube is in the shape of a hollow cylinder
and hollow hemisphere. Calculate the volume
of a liquid that can be held in the test tube.
18.5 cm
7.
4 3
πr .
3
Calculate the volume of a sphere of radius 1.7 cm, giving your answer correct to
1 decimal place.
The volume of a sphere of radius r is
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C4.3
Mathematics SKE, Strand C
8.
UNIT C4 Volumes: Text
A marble paperweight consists of a cuboid and a hemisphere as shown in the
diagram. The hemisphere has a radius of 4 cm.
4 cm
5 cm
10 cm
10 cm
Calculate the volume of the paperweight.
C4.4 Dimensions
Most quantities are based on three fundamental quantities, mass, length and time. The
letters M, L and T are used to describe the dimensions of these quantities.
If m is the mass of an object, then we write
[ m] = M
to show that M is the dimension of m.
If x is a length, then, similarly,
[ x] = L
[x ] = L
[x ] = L
2
2
3
3
If t is a time, then
[t ] = T
Here the emphasis will be on quantities involving lengths.
As areas are calculated by multiplying two lengths,
[Area ] = L2
or
Dimensions of Area = L2
[Volume] = L3
or
Dimensions of Volume = L3
Similarly,
The ideas of dimensions can be used to check formulae.
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Mathematics SKE, Strand C
C4.4
UNIT C4 Volumes: Text
Worked Example 1
The area of a circle is given by
A = π r2
(a)
What are the dimensions of A and r?
(b)
What are the dimensions of π?
Solution
(a)
A is an area, so
[ A] = L2
[ ]
r is a length, so [r ] = L and r 2 = L2
(b)
The dimension of the formula A = π r 2 must be consistent, so
[ A] = [π ] × [r 2 ]
But using the results from (a),
L2 = [π ] × L2
[π ] =
L2
= L0
L2
=1
The number π has no dimension.
Note
Numbers that are not lengths or other quantities have no dimension.
Worked Example 2
If x, y and z are all lengths, state whether each expression below could be for a length, an
area or a volume.
(a)
x+y+z
(b)
x yz
(d)
xy
z
(e)
x y + xz + yz
(c)
x
+y
z
Solution
(a)
[ x + y + z ] = [ x ] + [ y] + [ z ]
= L + L + L
As the formula contains 3 terms, all of dimension L, this expression gives a length.
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Mathematics SKE, Strand C
C4.4
(b)
[ x yz]
UNIT C4 Volumes: Text
= [ x ] × [ y] × [ z ]
= L × L × L
= L3
As the dimension of x yz is L3 this expression gives a volume.
(c)
[ x] + y = L + L
[] L
[z]
= 1 + L
The first term has no dimension and the second has a dimension, L, so the
expression is not dimensionally consistent.
(d)
[ x y]
[z]
=
[ x ] × [ y]
[z]
=
L×L
L
=
L2
L
= L
As the dimension of this quantity is L, this expression would give a length.
(e)
[ x y + xz + yz]
= [ x y] + [ xz ] + [ yz ]
= L2 + L2 +
L2
So the expression contains three terms, all of which have dimension L2, and so
could represent an area.
Challenge
A man wishes to take 4 litres of water out of a large tank of water, but he has only one
5-litre and one 3-litre jar. How can he do it?
Exercises
1.
If x, y and z all represent lengths, consider each expression and decide if it could be
for a length, an area, a volume or none of these.
(a) x y
(b)
x
y
(c)
x 2 y2
z
(d)
x3
z
(e)
x y2
z
(f)
x y
+
z
x
(g)
y2
+ y3
x
(h)
y z3
x
(i)
2x yz
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C4.4
Mathematics SKE, Strand C
2.
Which of the formulae listed below are not dimensionally consistent, if p, q and r
are lengths and A is an area?
(a) p = q + r
(d) A =
3.
UNIT C4 Volumes: Text
p2 q
r
(b)
p=
q
+r
r
(c)
A = pqr
(e)
p=
q2
r
(f)
q 2 = p2 + r 2
The diagram shows a triangular wedge.
Use dimensions to select the formula from
the list below that could give the surface
area of the wedge.
a
d
A = abc + bd + ac
c
A = ab + ac + bc + cd
b
A=a+b+c+d
A = abc + bcd + abd
4.
Consider sin θ =
opposite
hypotenuse
What are the dimensions of sin θ ?
5.
6.
The diagram shows a hole, which has a radius of r at the surface and a depth d.
Which of the following could give the volume of the hole?
V =
13
π rd
17
V =
13 2
πr d
17
V =
13 2
π rd
17
V =
13 3
πr d
17
r
d
Jared tries to find the volume of sellotape on a roll. Use dimensions to find the
formula from the list below that could be correct.
V = π ( R − r )h
r
V = π 2 Rh − π r 2 h
R
h
V = π Rh 2 − π rh
V = π R2 h − π r 2 h
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C4.4
UNIT C4 Volumes: Text
Mathematics SKE, Strand C
7.
Alex has forgotten the formula for the volume of a sphere. He writes down the
following formulae to try and jog his memory.
4 2
π r,
3
4 3
πr ,
3
3 2
π r,
4
4 2
πr ,
3
3 3
πr
4
(a)
Which of these formulae could give volumes?
(b)
Which of these formulae could give areas?
y
y
y
8.
Amil has the two expressions below; one is for
the volume of the shape and the other is for the
surface area.
6 x 2 + 4 y2
3
x +y
x
3
x
Explain what each expression gives.
9.
10.
x
Which of the formulae below could give the
volume of this shape?
V=
π dh π da
+
4
12
V =
π d 2 h π da
+
4
12
V =
π dh π da 2
+
4
12
V=
π d 2h π d 2a
+
4
12
a
h
d
(a)
If a, b and c are lengths and s =
(b)
If X =
a+b+c
, find the dimension of s.
2
s( s − a) ( s − b) ( s − c) , find the dimension of X and state what
type of quantity it could be.
11.
By considering dimensions, decide whether the following expressions could be a
formula for
perimeter, area, or volume.
In the expressions below, a, b and c are all lengths.
(a)
a+b+c
(b)
2 3
π a + π a2b
3
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UNIT C4 Volumes: Text
Mathematics SKE, Strand C
C4.4
12.
The expressions shown in the table below can be used to calculate lengths, areas or
volumes of various shapes.
π , 2, 4 and 12 are numbers which have no dimensions. The letters r, l, b and h
represent lengths.
Put a tick in the box underneath those expressions that can be used to calculate a
volume.
2π r
13.
4π r 2
π r 2h
π r2
lbh
1
2 bh
The diagram shows a child's play brick in
the shape of a prism.
The following formulae represent certain
quantities connected with this prism.
l
a
b
π ab, π (a + b), π abl π ( a + b)l
Which of these formulae represent areas?
14.
One of the formulae in the list below can be used to calculate the area of material
needed to make the curved surface of the lampshade in the diagram.
2
(i)
π h( a + b )
(iii)
π h( a + b ) ,
(ii)
π h 2 (a + b) ,
(iv)
π h 2 (a + b) .
a cm
h cm
2
b cm
State which formula is correct.
Give a reason for your answer.
Information
Archimedes (287BC-212BC), a Greek Mathematician, was once entrusted with the task of
finding out whether the King's crown was made of pure gold. While taking his bath, he
came up with a solution and was so excited that he dashed out into the street naked
shouting "Eureka" (I have found it). The container that you use in the Science laboratory
to measure the volume of an irregular object is known as an Eureka can (named after this
incident). Archimedes was so engrossed in his work that when his country was conquered
by the Romans, he was still working hard at his mathematics. When a Roman soldier
ordered him to leave his desk, Archimedes replied, "Don't disturb my circles." He was
killed by that soldier for disobeying orders.
Archimedes' greatest contribution was the discovery that the volume of a
2
sphere is that of a cylinder whose diameter is the same as the diameter of
3
the sphere. At his request, the sphere in the cylinder diagram was engraved
on his tombstone.
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