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Key words
Length, perimeter and area
millimetre (mm)
centimetre (cm)
metre (m)
kilometre (km)
perimeter
area
Know and use the names and abbreviations for units of length and area
Be able to measure and make a sensible estimate of length and area
Find out and use the formula for the perimeter of a rectangle and know
how to calculate the perimeter of shapes made from rectangles
Find out and use the formulae for the areas of a rectangle and of a right-angled triangle
Solve problems in everyday life involving length and area
Units of length are the millimetre (mm), centimetre (cm), metre (m) and
kilometre (km) .
10 mm 1 cm
1000 mm 100 cm 1 m
1 000 000 mm 100 000 cm 1000 m 1 km
Perimeter means the distance all the way around the boundary of a shape.
The perimeter of a rectangle is: 2 base 2 height
The area is the space inside a two-dimensional shape.
Units of area are the square millimetre (mm2), square centimetre (cm2), square metre (m2)
and square kilometre (km2) .
We can calculate the area of a rectangle by
multiplying the base by the height.
Area base height
We can calculate the area of a right-angled
triangle by imagining it is half of a rectangle.
1
Area 2 (base height)
height
height
base
base
Example
a) Find the perimeter of this shape.
b) Find the area of this shape.
A
B
7 cm
a) AB
10 cm
CD EF 6 cm 4 cm
10 cm
D
3 cm
F
Perimeter AB BC CD DE EF FA
6 cm
C
E
4 cm
10 cm 7 cm 6 cm 3 cm 4 cm 10 cm
40 cm
Find the length AB first.
b) Area of rectangle X 10 cm 7 cm
10 cm
A
70 cm2
X
Area of rectangle Y 3 cm 4 cm
12 cm2
D
Y
3 cm
Total area of shape 70 cm2 12 cm2 F
E
4 cm
82 cm2
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Maths Connect 1R
B
7 cm
C
Here the shape has been
broken down into two
rectangles to find the area.
An alternative method
would be to subtract the
area of the rectangle that is
missing from the corner of
the large rectangle.
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Exercise 3.1 .............................................................................................
What units would you use to measure a) the perimeters b) the areas of the following shapes:
Choose from: mm, cm, m, km, mm2, cm2, m2 and km2.
i) the sole of your foot
ii) the cover of this book
iii) the top of the table
iv) the floor of your bedroom
v) a mouse’s footprint
vi) Spain.
Change all the
measurements to mm first.
Calculate the perimeters of these shapes.
9m
a)
3m
b) A
3m
6m
6m
c) A
B
B
2 cm
1m
3m
D
2m
C
F
1m C
15 m
F
D
1.3 cm
30 mm E
E
32 mm
H
4.1 cm
G
These shapes are made from rectangles. Find their areas.
a)
5m
10 cm
b)
c)
1m
1m
6m
1m
2m
1 cm
1 cm
1 cm
1m
7m
1m
15 m
10 cm
1 cm
Find the areas of these right-angled triangles:
a)
b)
8 cm
c)
19 mm
5 cm
6 cm
10 cm
6.3 cm
4 cm
Find the base and height of rectangles which have:
a) area of 10 cm2, perimeter 14 cm
b) area of 24 cm2, perimeter 22 cm
c) area of 16 mm2, perimeter 16 mm
d) area of 7.5 km2, perimeter 13km.
Explain how you worked these out.
You might find it helpful to sketch
the rectangles. Remember that a
square is also a type of rectangle.
Investigation
Use squared paper.
a) How many different rectangles can you draw using 12 whole squares?
b) Do they all have the same perimeter?
c) Do they all have the same area?
Make other shapes using 12 whole squares and calculate their perimeters and areas.
d) What do you notice?
e) Write a hint for someone who is trying to make a rectangle from 100 squares with
the smallest possible perimeter.
Length, perimeter and area 27
03 Section 3 pp026-033.qxd
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Key words
Area of composite shapes
square centimetre (cm2)
square metre (m2)
square millimetre (mm2)
square kilometre (km2)
area
Know how to calculate the areas of a triangle, a parallelogram
and a trapezium
The area is the space inside a two-dimensional shape.
The formula for the area of a rectangle is base height
1
The formula for the area of a right-angled triangle is 2 (base height)
Shapes that do not have right angles
have slant heights as well as
perpendicular heights. We need to
know the perpendicular height of the
shape to find the area.
slant
height
perpendicular
height
We can calculate the area of a
parallelogram by doubling the area of a
non right-angled triangle.
Area base (perpendicular) height
These
triangles
have the
same area.
perpendicular
height
base
We can calculate the area of a non- rightangled triangle by splitting it into two
right-angled triangles, or by imagining it
is half a rectangle.
We can find the area of a trapezium by
breaking it into a rectangle and triangles
and then calculating the areas of these
shapes.
1
Area 2 (base perpendicular height)
perpendicular
height
base
Example
These
triangles
have the
same area.
6 cm
Find the area of this shape.
4 cm
Area of parallelogram base perpendicular height
10 cm
6 cm 4 cm
24 cm2
Area of triangle
1
2 (base perpendicular height)
1
2 6 (10 4)
1
2 36
18 cm2
Total area
24 cm2 18 cm2
42 cm2
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Maths Connect 1R
The shape can be broken
down into a parallelogram
and a triangle.
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Exercise 3.2 .............................................................................................
Calculate the areas of these triangles:
a) 5 mm
b)
c)
4 cm
32 mm
15 cm
43 mm
5 cm
Calculate the areas of the following shapes:
a)
b)
c)
6 cm
d)
5 cm
6 cm
7 cm
6 cm
15 m
4m
8 cm
8 cm
2 cm
7m
10 cm
2 cm
The shaded shapes on this grid can be broken down into simpler shapes. Calculate the
areas, in square units, of each of the shaded shapes.
a)
b)
c)
Here are two trapeziums with parallel sides of length 5 cm and 3 cm and a perpendicular
height of 2 cm:
3 cm
3 cm
2 cm
2 cm
1 cm
1 cm
5 cm
0.5 cm
1.5 cm
5 cm
a) Find the areas of the two trapeziums.
b) Draw two more trapeziums with the same dimensions and find their areas. What do
you notice?
Investigation
Albert has a rabbit and 12 m of fencing. He wants to make a run for the rabbit.
Albert thinks that the area enclosed by the fencing is always the same, regardless of
the shape of the run. Show why Albert is wrong. What is the largest area he could
enclose? Show how you know.
Area of composite shapes 29
03 Section 3 pp026-033.qxd
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Key words
Connecting 2-D and 3-D
face
edge
vertex
vertices
cube
cuboid
3-D
Use other 2-D shapes to visualise and describe 3-D shapes and
consider their properties
Be able to draw 2-D representations of 3-D shapes
A face is the flat surface of a solid.
An edge is where two faces meet.
edge
A vertex is where three or more edges meet.
vertex
Vertices is the plural of vertex.
face
A cube has six identical square faces.
A cuboid has three pairs of rectangular faces.
Opposite faces are the same shape and size.
A tetrahedron has four triangular faces.
cube
cuboid
tetrahedron
A prism has a uniform cross section.
A hemisphere has one circular face and one
curved face.
A cylinder has two circular faces and one
curved face.
triangular
prism
We sometimes use isometric paper to draw
representations of 3-D shapes.
cube
Example 1
Describe this shape.
This shape is a prism. It has two L-shaped faces and six
rectangular faces. It has 18 edges and 12 vertices.
Example 2
Draw this shape on isometric paper.
Draw all of the front faces
first, then the side faces
and finally the top faces.
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Maths Connect 1R
hemisphere
cylinder
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Exercise 3.3 .............................................................................................
Work with a partner. Take a handful of cubes each. Take it in turns to make a 3-D shape
and describe it so that your partner can make it. If your cubes are coloured, try not to use
colour as a clue!
Use isometric paper to draw these shapes made from cubes. Shade the front faces.
a)
b)
c)
d)
Draw all the front faces first.
Build a shape game. You will need a normal dice and some isometric paper.
Take turns to throw the dice. 1 square; 2 rectangle; 3 triangle; 4 circle;
5 hexagon; 6 another throw. Collect a face for each number you throw. The winner is
the first to collect enough 2-D faces to make a 3-D shape and to sketch that shape.
This skeleton 3-D shape is made from four 10 cm lengths of
5 cm
wire and eight 5 cm lengths of wire.
5 cm
Write the list of lengths of wire for each of the skeletons
shown below:
N
M
5 cm
4 cm
10 cm
O
8 cm
5 cm
4 cm
P
10 cm
5 cm
4 cm
Q
R
10 cm
10 cm
12 cm
8 cm
10 cm
4 cm
6 cm
6 cm
10 cm
8 cm
a) For each of the shapes in Q4, record the number of vertices, face and edges.
Can you find the connection between the number of vertices, faces and edges
in each shape?
b) How many edges would a shape with eight faces and
Try adding the number of
twelve vertices have? Sketch a shape like this on isometric
faces and vertices together.
paper.
Investigation
There is only one possible model that can be made from two cubes.
Do not count
rotations of the
same model.
There are only two possible models that can be made from three cubes.
Make models from four and five cubes and draw them on isometric
paper. How can you be certain that you have made all the possible models?
Connecting 2-D and 3-D 31
03 Section 3 pp026-033.qxd
3.4
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Key words
Cubes and cuboids
surface area
net
cube
cuboid
Investigate the nets of cubes and cuboids
Know and use a formula for the surface area of a cube and cuboid
Solve simple problems involving lengths, perimeters and surface areas
of shapes made from cubes and cuboids
The surface area of a 3-D shape is the total area of all its faces.
Drawing the net helps us to make sure we have added the areas of all of the faces.
The surface area of a cube of side 5 cm is the
sum of the area of its six faces. This is
w
w
calculated as 6 (5 5) cm2.
w
The general formula for the surface area
of a cube of side w is 6 ( w w).
w
w
w
w
w
w
The surface area of a cuboid of width 3 cm,
length 4 cm and height 5 cm is the sum of
the areas of its three pairs of faces.
h
w
This is calculated as
2 (3 4) 2 (3 5) 2 (4 5) cm2.
w
l
The general formula for the surface area of a
cuboid of width w, length l and height h is
2 (l w) 2 (l h) 2 (w h).
w
w
w
l
h
h
h
Example
Find the surface area of
these shapes.
a)
b)
3 cm
4 cm
10 cm
8 cm
3 cm
6 cm
3 cm
a) Area of purple rectangle 6 10 cm 60 cm2
Area of pink rectangle
3 10 cm 30 cm2
Area of brown rectangle 6 3 cm 18 cm2
Total surface area
2 60 2 30 2 18 cm2
216 cm2
b) Area of yellow face 8 3 3 4 36
6 cm
4 cm
3 cm
Alternatively, you can
use the formula for
surface area, given in
the explanation box.
Replace w, l and h with
the width, length and
height of the cuboid.
but there are 2 of these so area of yellow faces 72
Area of the 2 green faces 4 3 4 3 24
Area of the 2 blue faces 3 3 3 3 18
Area of the base 6 3 18
Area of the back 8 3 24
Total surface area 72 24 18 18 24 156 square units
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Maths Connect 1R
You may find it helpful
to sketch the different
faces of the shape with
their dimensions.
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Exercise 3.4 .............................................................................................
Here are the nets of different cuboids. For each one, find the area of the yellow, blue and
green faces. Then find the total surface area for each net.
a)
3 mm
10 mm
3 mm
5 mm
b)
3 cm
d)
0.1 m
5 cm
3 cm
5 mm
2m
5 cm
3 mm
3 cm
5 mm
1m
c)
70cm
3 cm
5 cm
3 mm
70 cm
0.1 m
2m
5 cm
3 cm
1m
Find the surface areas of each of the following shapes:
a)
b)
3 cm
3 cm
3 cm
c)
d)
10 cm
10 cm
15 cm
10 cm
30 cm
3 cm
3 cm
10 cm
These shapes are made from cuboids.
a) Find the areas of the front
faces.
b) Find the total surface areas
of the shapes.
5 cm
4 cm
2 cm
5 cm
4 cm
1 cm
8 cm
7 cm
1 cm
10 cm
6 cm
1 cm
9 cm
Mr Harding is building a rectangular pond for his garden.
The pond is 1 m deep. The length is 2 m and the width is 2.5 m.
He wants to line the sides and bottom of his pond with tiles
that are 25 cm square. How many tiles will he need?
Investigation
You have two of each of the following rectangles:
Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle
1
2
3
4
5
6
7
8
Length (cm)
3
3
4
5
3
2
1
1
Width (cm)
5
4
5
6
6
3
2
3
Investigate all the different cuboids that can be made using these rectangles as faces.
Keep a record of the length, width, height and surface area of the different cuboids.
What do you notice?
Cubes and cuboids 33