SHAPE, SPACE AND MEASURES
Volume
level
level
5
Volume of a cuboid
6
Capacity
• Capacity is the amount of space inside a hollow 3-D shape.
• Capacity usually refers to the volume of a gas or liquid. You need to know
1000 cm3 = 1 litre.
50 cm
Example: Find the volume of this fish tank,
40 cm
giving your answer in litres.
• Volume is the amount of space inside a 3-D shape.
• The common units for volume are: mm3, cm3 or m3.
V = 50 x 40 x 30
= 60 000 cm3
• Volume = length x width x height
height
30 cm
V = 60 litres
V=lxwxh
V = lwh
Example: This is a net of a cuboid. If one
width
square has an area of 1 cm2,
what is the volume of the cuboid?
length
Example: Find the volume of this cuboid.
V = lwh
= 12 x 3 x 5 = 180 cm3.
5 cm
Top Tip!
Sample mental
Substitute numbers into a
formula before trying to
work anything out.
3 cm
12 cm
N
ot
l
r
o
f
Top Tip!
You should know these
cube roots:
3
1=1
3
8=2
3
27 = 3
3
64 = 4
3
125 = 5
3
1000 = 10
The volume of a cube is 27 cm3.
What is the length of an edge of the cube?
Since 27 = 3 x 3 x 3, the length of an edge = 3 cm.
6
w
level
h
• The surface area is given by
test question
level
Surface area of a cuboid
• There are 6 faces on a cuboid, with opposite
faces having the same area.
b
i
r
t
dis
n
o
uti
V = 4 x 3 x 2 = 24 cm3
Sample National
6
Test question
A = 2lw + 2lh + 2wh
cuboid in the panel above.
A = 2 x 12 x 3 + 2 x 12 x 5 + 2 x 3 x 5 = 72 + 120 + 30 = 222
pot Check
3 cm
These two cuboids have the same volume.
Find the value of x.
Example: Find the surface area of the purple
1 What is the volume and surface
area of this cuboid?
66
3 cm
3 cm
2 cm
cm2.
4 cm
4 cm
Answer
Volume of first cuboid = 36 cm3.
So volume of second cuboid = 6x = 36 cm3.
So x = 6 cm.
3 cm
2 cm
x
can
of the Sun
e
m
lu
o
v
e
Th
arths.
a million E
hold over
67
SHAPE, SPACE AND MEASURES
Constructions
Angle bisector
Constructing triangles
level
• When constructing triangles it is very important that you measure lines and
angles accurately.
A
Part of the diagram, such as
the base line, is often drawn
for you. Always use a compass
to mark out the distances
rather than a ruler.
7 cm
8 cm
t
o
N
C
r
o
f
Perpendicular bisector
level
• The perpendicular bisector is the line that passes through the midpoint of two
other points and is perpendicular (at right angles) to the line that joins them.
A
B
First draw a line 8 cm long.
Then use a compass to measure 4 cm and draw an arc from the left-hand
end of the 8 cm line.
Then use a compass to measure 7 cm and draw an arc from the right-hand end
of the 8 cm line.
Then join the ends of the 8 cm line to the point where the arcs cross.
7
b
i
r
t
dis
First set the compass to about 3 cm.
From B, draw arcs on BA and BC.
Where these arcs cross BA and BC, draw two
B
further arcs to cross each other.
Draw a line from B through the point where these arcs cross.
This is the angle bisector of ABC.
n
o
uti
level
Example: Draw an angle of 60° at the point B.
First set the compass to about two-thirds of
the distance from A to B.
Draw arcs from A on both sides of the line.
Without changing the size of the compass,
do the same from B.
Join the points where the arcs cross.
This line is the perpendicular bisector of AB.
Test question
Construct a triangle that has the
following properties:
A
Top Tip!
Make sure your arcs are shown.
B
7
Set the compass to about 3 cm.
Draw an arc from B that crosses the line and
draws almost a quarter circle.
From the point where the arc crosses the line,
draw another arc to cross the first.
Join the point where these arcs cross to B.
The angle at B is 60°.
Sample National
B
C
Constructing an angle of 60°
Example: Construct the perpendicular bisector of AB.
A
7
Example: Construct the angle bisector of ABC.
Top Tip!
Example: Construct this triangle accurately.
4 cm
6
level
• The angle bisector is the line that passes at the same distance from
two intersecting lines.
level
7
• Total length of three sides is 12 cm.
• Only two of the sides are equal length.
• All sides are whole numbers of
5 cm
centimetres.
Answer
There is only one possible answer.
A triangle with sides of 2 cm, 5 cm
and 5 cm.
5 cm
B
ylonians
o, the Bab
g
a
rs
a
e
y
0
ss the
About 400
e Sun acro
th
f
o
th
a
t 360
ep
year (abou
tracked th
a
k
o
to
it
led
alised
ircuit. This
c
e
sky and re
n
o
te
omple
360°.
days) to c
e circle into
th
e
id
iv
d
them to
pot Check
1 Draw this triangle accurately.
5 cm
2 cm
35°
6 cm
68
69
SHAPE, SPACE AND MEASURES
Loci
level
8
Loci
level
7
• Most loci problems are set in a real-life context.
Paths
Example: A radio transmitter is to be built so that it is the same distance from two
• A locus (singular of loci) is the path moved by a point according to a rule.
towns: Radville and Seeton. It also has to be within 20 km of a third town,
Towton. Show the possible location of the transmitter.
‘Same distance from’ means ‘the perpendicular bisector of’.
‘Within 20 km’ means inside a circle of radius 20 km.
The ‘overlap’ of these two conditions is shown with the red line.
Example: Draw the locus of all the points that are
a exactly 2 cm from A
A
B
b within 3 cm of B.
a Points that are exactly 2 cm from A form a circle of radius
2 cm centred on A.
b Points that are within 3 cm of B are all points inside a circle
of radius 3 cm centred on B.
Transmitter could
be positioned
anywhere on this
red line
Radville
Towtown
a
b
n
o
ti
B
A
Example: Draw the locus of all points that are
a exactly 2 cm from the line AB
b the same distance from A as from B.
A
t
o
N
a The points that are exactly 2 cm from AB form a ‘sausage’ shape
around AB with two straight lines 2 cm away each side and two
semi-circles of radius 2 cm centred on A and B.
b The points that are the same distance from A and B are the points on
the perpendicular bisector of AB.
b
B
r
o
f
u
b
i
r
t
s
di
Sample National
Seeton
level
70
8
Test question
The plan shows a garden. Each square is 1 m by 1 m.
There are four trees in the garden whose trunks are marked by T.
John wants to erect an aerial for his short wave radio.
The aerial cannot be
• within 2 metres of any tree trunk
• nearer than 1 metre to the edge of the garden.
Show the places where the aerial could be placed.
Answer
A circle of radius 2 m must be drawn round each tree
and all the area within 1 metre of the edge must be
excluded. The prohibited areas are shaded.
The area that is unshaded is where the
aerial could be erected.
B
Make sure your construction
arcs are shown and that the
required locus is clearly marked.
Scale: 1 cm represents 10 km
a
A
Top Tip!
the
rth around
a
E
e
th
f
o
elliptical.
The locus
cular but
ir
c
t
o
n
is
Sun
71
T
T
T
T
T
T
T
T
SHAPE, SPACE AND MEASURES
Volume
levels
5-6
levels
5-6
1
7 Here are four cuboids.
a What is the volume of this cuboid?
4 cm
1 mark
5 cm
cm3
b What is the surface area?
A
18 cm
3 cm
C
D
12 cm
5 cm
3 cm
12 cm
5 cm 5 cm
1 mark
1 cm
5 cm
B
4 cm
cm2
4 cm
25 cm
Rearrange the cuboids in the order
of their volume, with the smallest first.
cm3.
2 A cuboid has a volume of 36
Its length is 6 cm and its width is 3 cm.
What is the height of the cuboid?
cm
3 A cuboid has a volume of 200 cm3.
Its length and width are 5 cm.
What is the surface area?
Remember to include the units in your answer.
1 mark
r
o
tf
o
N
b
i
r
t
dis
n
o
uti
8 A tank has the following measurements.
How many litres of water can it hold?
50 cm
2 marks
4 The volume of a cube is 64 cm3.
What is the length of each edge of the cube?
cm
5 The surface area of this cuboid is 184 cm2.
Work out the length of the cuboid.
1 mark
80 cm
2m
1 mark
litres
2 marks
cm
1 mark
9 These two cuboids have the same volume. What is the value of x?
4 cm
5 cm
cm
1 mark
3 cm
2 cm
2m
6 This is a net of a cuboid.
What is the volume of the cuboid?
3 cm
8 cm
4 cm
x
3m
m3
1m
66
1 mark
67
SHAPE, SPACE AND MEASURES
Constructions
levels
6-7
levels
6-7
5 Draw this triangle accurately.
1 Construct an angle of 60° at the point A on the line AB.
7 cm
40°
8 cm
A
B
2 marks
2 marks
2 Construct the perpendicular bisector of the line AB.
A
B
r
o
tf
o
N
2 marks
b
i
r
t
dis
n
o
uti
6 Draw this triangle accurately.
3 Construct the angle bisector of the angle ABC.
50°
A
35°
8 cm
2 marks
2 marks
B
C
4 Draw this triangle accurately.
7 cm
4 cm
6 cm
2 marks
68
69
SHAPE, SPACE AND MEASURES
Loci
levels
7-8
levels
7-8
1 In each of these squares shade the region described.
a
P
Q
b
P
Q
4 The diagram shows an island with two airports A and B.
The scale is 1cm represents 10 km.
A radar station at A picks up aircraft within 30 km.
A radar station at B picks up aircraft within 40 km.
A
S
R
S
A
R
a All points that are nearer to P than to Q.
1 mark
b All points that are nearer to S than to Q.
1 mark
2 ABCD are squares of side 3 cm. Match the given loci to the diagrams.
A
B
Loci a
B
D
C
D
C
A
B
A
B
Loci d
D
C
b
i
r
t
dis
n
o
uti
a Does the radar station at B pick up an aircraft flying directly over A?
Loci b
Loci c
i
A
B B
D
r
o
tf
o
N
b Show all the points where aircraft are picked up by both radar stations.
2 marks
5 The diagram shows a garden with a garden shed.
Each grid square represents 50 cm.
C
All points nearer to D than to B.
ii All points within 3 cm of D.
iii All points nearer to the line AD than the line BC.
iv All points within 2 cm of B.
4 marks
Shed
3 Construct the locus of the point that is the same distance from the lines AB and AC.
B
2 marks
A
A tree is to be planted. It must not be planted within 1 m of
the edge of the garden or the shed.
Shade clearly the area in which the tree can be planted.
C
70
71
1 mark
MATHS WORKBOOK 5–8
Shape, space and
measures answers
Pages 66–67 Volume
1 a 15 cm3
b 46 cm2
2 2 cm
3 210 cm2 (1 mark for units)
4 4 cm
5 8 cm
6 6 m3
7 D = 180 cm3,
C = 240 cm3, A = 288 cm3,
B = 625 cm3
8 800 l (1 mark for 800 000 cm3 or 0.8 m3)
9 4 cm
Pages 68–69 Constructions
1 (1 mark for arcs, 1 for accuracy)
2 (1 mark for arcs, 1 for accuracy)
3 (1 mark for arcs, 1 for accuracy)
r
o
tf
b
i
r
t
dis
4 (1 mark for 2 correct sides, 1 mark for all correct)
5 (1 mark for 1 correct side and 1 angle, 1 mark for all correct)
o
N
6 (1 mark for 1 correct side and 1 angle, 1 mark for all correct)
1
n
o
uti
Shape, space and
measures answers
MATHS WORKBOOK 5–8
Pages 70–71 Loci
1 a
P
Q
S
R
2 i b
ii c
b
Q
S
R
iii a iv d
B
3
P
(1 mark for arcs)
A
C
4 a No
b Shown half scale
A
B
5
o
N
r
o
tf
Shed
b
i
r
t
dis
2
n
o
uti