Transcript Document

KS3 Mathematics
S8 Perimeter, area and
volume
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© Boardworks Ltd 2004
Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Put these shapes in order
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Perimeter
To find the perimeter of a shape we add together the
length of all the sides.
What is the perimeter of this shape?
Starting point
Perimeter = 3 + 3 + 2 + 1 + 1 + 2
3
= 12 cm
2
3
1
1
2
1 cm
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Perimeter of a rectangle
To calculate the perimeter of a rectangle we can use a
formula.
length, l
width, w
Using l for length and w for width,
Perimeter of a rectangle = l + w + l + w
= 2l + 2w
or
= 2(l + w)
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Perimeter
Sometime we are not given the lengths of all the sides.
We have to work them out from the information we are
given.
9 cm
For example, what is the
perimeter of this shape?
5 cm
The lengths of two of
the sides are not given
12 cm
4 cm
so we have to work
them out before we can
a cm
find the perimeter.
Let’s call the lengths a and b.
b cm
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Perimeter
Sometime we are not given the lengths of all the sides.
We have to work them out from the information we are
given.
9 cm
a = 12 – 5
= 7 cm
5 cm
b=9–4
= 5 cm
12 cm
4 cm
a cm
7
P = 9 + 5 + 4 + 7 + 5 + 12
= 42 cm
b cm
5
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Perimeter
Calculate the lengths of the missing
sides to find the perimeter.
5 cm
p = 2 cm
2 cm
p
q
r
s
t = 2 cm
u = 10 cm
4 cm
2 cm
u
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s = 6 cm
6 cm
4 cm
t
q = r = 1.5 cm
2 cm
P = 5 + 2 + 1.5 + 6 + 4
+ 2 + 10 + 2 + 4 + 6
+ 1.5 + 2
= 46 cm
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Perimeter
What is the perimeter of this shape?
5 cm
4 cm
Remember, the dashes
indicate the sides that are
the same length.
P=5+4+4+5+4+4
= 22 cm
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Perimeter
What is the perimeter of this shape?
Start by finding the lengths
of all the sides.
4.5 cm
4.5 m
5m
Perimeter = 4.5 + 2 + 1 + 2
+ 1 + 2 + 4.5
4m
= 17 cm
2m
1 cm
2m
1 cm
2m
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Perimeter
What is the perimeter of this shape?
Before we can find the
perimeter we must
convert all the lengths to
the same units.
256 cm
3003cm
m
1.9 m
190
cm
In this example, we can
either use metres or
centimetres.
Using centimetres,
2.4 m
240
cm
P = 256 + 190 + 240 + 300
= 986 cm
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Equal perimeters
Which shape has a different
perimeter from the first shape?
A
B
A
B
C
B
C
A
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C
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Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Area
The area of a shape is a measure of how much surface
the shape takes up.
For example, which of these rugs covers a larger surface?
Rug A
Rug C
Rug B
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Area of a rectangle
Area is measured in square units.
For example, we can use mm2, cm2, m2 or km2.
The 2 tells us that there are two dimensions, length and width.
We can find the area of a rectangle by multiplying the length
and the width of the rectangle together.
length, l
width, w
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Area of a rectangle
= length × width
= lw
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Area of a rectangle
What is the area of this rectangle?
4 cm
8 cm
Area of a rectangle = lw
= 8 cm × 4 cm
= 32 cm2
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Area of a right-angled triangle
What proportion of this rectangle has been shaded?
4 cm
8 cm
What is the shape of the shaded part?
What is the area of this right-angled triangle?
1
Area of the triangle =
× 8 × 4 = 4 × 4 = 16 cm2
2
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Area of a right-angled triangle
We can use a formula to find the area of a right-angled
triangle:
height, h
base, b
Area of a triangle =
1
× base × height
2
1
bh
=
2
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Area of a right-angled triangle
Calculate the area of this right-angled triangle.
To work out the area of
this triangle we only need
the length of the base
and the height.
8 cm
10 cm
6 cm
We can ignore the third
length opposite the right
angle.
1
× base × height
2
1
=
× 8 × 6 = 24 cm2
2
Area =
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Area of shapes made from rectangles
How can we find the area of this shape?
We can think of this shape
as being made up of two
rectangles.
7m
A
Either like this …
10 m
… or like this.
15 m
8m
B
15 m
Label the rectangles A and B.
5m
Area A = 10 × 7 = 70 m2
Area B = 5 × 15 = 75 m2
Total area = 70 + 75 = 145 m2
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Area of shapes made from rectangles
How can we find the area of the shaded shape?
7 cm
A
8 cm
4 cm
3 cm
B
We can think of this shape
as being made up of one
rectangle cut out of another
rectangle.
Label the rectangles A and B.
Area A = 7 × 8 = 56 cm2
Area B = 3 × 4 = 12 cm2
Total area = 56 – 12 = 44 cm2
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Area of an irregular shapes on a pegboard
How can we find the area of this irregular
quadrilateral constructed on a pegboard?
We can divide the shape into
right-angled triangles and a
square.
Area A = ½ × 2 × 3 = 3 units2
A
D
E
C
B
Area B = ½ × 2 × 4 = 4 units2
Area C = ½ × 1 × 3 = 1.5 units2
Area D = ½ × 1 × 2 = 1 unit2
Area E = 1 unit2
Total shaded area = 11.5 units2
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Area of an irregular shapes on a pegboard
How can we find the area of this irregular
quadrilateral constructed on a pegboard?
A
D
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B
C
An alternative method would
be to construct a rectangle
that passes through each of
the vertices.
The area of this rectangle is
4 × 5 = 20 units2
The area of the irregular
quadrilateral is found by
subtracting the area of each
of these triangles.
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Area of an irregular shapes on a pegboard
How can we find the area of this irregular
quadrilateral constructed on a pegboard?
A
C
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B
D
Area A = ½ × 2 × 3 = 3 units2
Area B = ½ × 2 × 4 = 4 units2
Area C = ½ × 1 × 2 = 1 units2
Area D = ½ × 1 × 3 = 1.5 units2
Total shaded area = 9.5 units2
Area of irregular quadrilateral
= (20 – 9.5) units2
= 11.5 units2
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Area of an irregular shape on a pegboard
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Area of a triangle
What proportion of this rectangle has been shaded?
4 cm
8 cm
Drawing a line here might help.
What is the area of this triangle?
1
Area of the triangle =
× 8 × 4 = 4 × 4 = 16 cm2
2
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Area of a triangle
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Area of a triangle
The area of any triangle can be found using the formula:
1
Area of a triangle =
× base × perpendicular height
2
perpendicular height
base
Or using letter symbols,
1
Area of a triangle =
bh
2
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Area of a triangle
What is the area of this triangle?
6 cm
7 cm
Area of a triangle =
=
1
bh
2
1
×7×6
2
= 21 cm2
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Area of a parallelogram
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Area of a parallelogram
The area of any parallelogram can be found using the formula:
Area of a parallelogram = base × perpendicular height
perpendicular
height
base
Or using letter symbols,
Area of a parallelogram = bh
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Area of a parallelogram
What is the area of this parallelogram?
8 cm
7 cm
We can ignore
this length
12 cm
Area of a parallelogram = bh
= 7 × 12
= 84 cm2
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Area of a trapezium
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Area of a trapezium
The area of any trapezium can be found using the formula:
1
Area of a trapezium = (sum of parallel sides) × height
2
a
perpendicular
height
b
Or using letter symbols,
1
Area of a trapezium =
(a + b)h
2
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Area of a trapezium
What is the area of this trapezium?
1
Area of a trapezium =
(a + b)h
2
6m
9m
=
1
(6 + 14) × 9
2
1
=
× 20 × 9
2
14 m
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= 90 m2
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Area of a trapezium
What is the area of this trapezium?
1
Area of a trapezium =
(a + b)h
2
=
8m
3m
12 m
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1
(8 + 3) × 12
2
1
=
× 11 × 12
2
= 66 m2
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Area problems
This diagram shows a yellow square inside a blue square.
What is the area of the
yellow square?
3 cm
7 cm
10 cm
We can work this out by subtracting
the area of the four blue triangles from
the area of the whole blue square.
If the height of each blue triangle is 7 cm, then the base is 3 cm.
Area of each blue triangle = ½ × 7 × 3
= ½ × 21
= 10.5 cm2
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Area problems
This diagram shows a yellow square inside a blue square.
What is the area of the
yellow square?
3 cm
7 cm
10 cm
We can work this out by subtracting
the area of the four blue triangles from
the area of the whole blue square.
There are four blue triangles so,
Area of four triangles = 4 × 10.5 = 42 cm2
Area of blue square = 10 × 10 = 100 cm2
Area of yellow square = 100 – 42 = 58 cm2
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Area formulae of 2-D shapes
You should know the following formulae:
h
Area of a triangle =
1
bh
2
b
h
Area of a parallelogram = bh
b
a
h
Area of a trapezium =
1
(a + b)h
2
b
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Using units in formulae
Remember, when using formulae we must make sure that all
values are written in the same units.
For example, find the area of this trapezium.
76 cm
518 mm
Let’s write all the lengths in cm.
518 mm = 51.8 cm
1.24 m = 124 cm
1.24 m
Area of the trapezium = ½(76 + 124) × 51.8 Don’t forget to
= ½ × 200 × 51.8
put the units at
the end.
= 5180 cm2
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Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The top and the bottom of the
cuboid have the same area.
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Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The front and the back of the
cuboid have the same area.
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Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The left hand side and the right
hand side of the cuboid have
the same area.
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Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
Can you work out the
5 cm
surface area of this cubiod?
8 cm
The area of the top = 8 × 5
= 40 cm2
7 cm
The area of the front = 7 × 5
= 35 cm2
The area of the side = 7 × 8
= 56 cm2
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Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
8 cm
5 cm
So the total surface area =
2 × 40 cm2
7 cm
Top and bottom
+ 2 × 35 cm2 Front and back
+ 2 × 56 cm2 Left and right side
= 80 + 70 + 112 = 262 cm2
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Formula for the surface area of a cuboid
We can find the formula for the surface area of a cuboid
as follows.
Surface area of a cuboid =
l
h
w
2 × lw
Top and bottom
+ 2 × hw
Front and back
+ 2 × lh
Left and right side
= 2lw + 2hw + 2lh
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Surface area of a cube
How can we find the surface area of a cube of length x?
All six faces of a cube have the
same area.
The area of each face is x × x = x2
Therefore,
x
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Surface area of a cube = 6x2
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Chequered cuboid problem
This cuboid is made from alternate purple and green
centimetre cubes.
What is its surface area?
Surface area
=2×3×4+2×3×5+2×4×5
= 24 + 30 + 40
= 94 cm2
How much of the
surface area is green?
48 cm2
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Surface area of a prism
What is the surface area of this L-shaped prism?
3 cm
3 cm
4 cm
6 cm
To find the surface area of
this shape we need to add
together the area of the two
L-shapes and the area of the
6 rectangles that make up
the surface of the shape.
Total surface area
5 cm
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= 2 × 22 + 18 + 9 + 12 + 6
+ 6 + 15
= 110 cm2
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Using nets to find surface area
It can be helpful to use the net of a 3-D shape to calculate its
surface area.
Here is the net of a 3 cm by 5 cm by 6 cm cubiod.
6 cm
3 cm
18 cm2
3 cm
5 cm 15 cm2
30 cm2
15 cm2
3 cm
18 cm2
3 cm
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6 cm
30 cm2
Write down the
area of each
face.
Then add the
areas together
to find the
surface area.
Surface Area = 126 cm2
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Using nets to find surface area
Here is the net of a regular tetrahedron.
What is its surface area?
Area of each face = ½bh
= ½ × 6 × 5.2
= 15.6 cm2
Surface area = 4 × 15.6
5.2 cm
= 62.4 cm2
6 cm
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Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Making cuboids
The following cuboid is made out of interlocking cubes.
How many cubes does it contain?
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Making cuboids
We can work this out by dividing the cuboid into layers.
The number of cubes in each layer
can be found by multiplying the
number of cubes along the length
by the number of cubes along the
width.
3 × 4 = 12 cubes in each layer
There are three layers altogether
so the total number of cubes in the
cuboid = 3 × 12 = 36 cubes
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Making cuboids
The amount of space that a three-dimensional object takes
up is called its volume.
Volume is measured in cubic units.
For example, we can use mm3, cm3, m3 or km3.
The 3 tells us that there are three dimensions, length, width
and height.
Liquid volume or capacity is measured in ml, l, pints or
gallons.
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Volume of a cuboid
We can find the volume of a cuboid by multiplying the area of
the base by the height.
The area of the base
= length × width
So,
height, h
Volume of a cuboid
= length × width × height
= lwh
width, w
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length, l
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Volume of a cuboid
What is the volume of this cuboid?
Volume of cuboid
= length × width × height
5 cm
= 5 × 8 × 13
8 cm
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13 cm
= 520 cm3
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Volume and displacement
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Volume and displacement
By dropping cubes and cuboids into a measuring cylinder
half filled with water we can see the connection between the
volume of the shape and the volume of the water displaced.
1 ml of water has a volume of 1 cm3
For example, if an object is dropped into a measuring
cylinder and displaces 5 ml of water then the volume of the
object is 5 cm3.
What is the volume of 1 litre of water?
1 litre of water has a volume of 1000 cm3.
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Volume of a prism made from cuboids
What is the volume of this L-shaped prism?
3 cm
We can think of the shape as
two cuboids joined together.
3 cm
4 cm
Volume of the green cuboid
= 6 × 3 × 3 = 54 cm3
6 cm
Volume of the blue cuboid
= 3 × 2 × 2 = 12 cm3
Total volume
5 cm
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= 54 + 12 = 66 cm3
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Volume of a prism
Remember, a prism is a 3-D shape with the same
cross-section throughout its length.
3 cm
We can think of this prism as lots
of L-shaped surfaces running
along the length of the shape.
Volume of a prism
= area of cross-section × length
If the cross-section has an area
of 22 cm2 and the length is 3 cm,
Volume of L-shaped prism = 22 × 3 = 66 cm3
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Volume of a prism
What is the volume of this triangular prism?
7.2 cm
4 cm
5 cm
Area of cross-section = ½ × 5 × 4 = 10 cm2
Volume of prism = 10 × 7.2 = 72 cm3
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Volume of a prism
What is the volume of this prism?
12 m
7m
4m
3m
5m
Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 cm2
Volume of prism = 5 × 72 = 360 m3
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Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Circle circumference and diameter
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The value of π
For any circle the circumference is always just over
three times bigger than the radius.
The exact number is called π (pi).
We use the symbol π because the number cannot be written
exactly.
π = 3.141592653589793238462643383279502884197169
39937510582097494459230781640628620899862803482
53421170679821480865132823066470938446095505822
31725359408128481117450284102701938521105559644
62294895493038196 (to 200 decimal places)!
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Approximations for the value of π
When we are doing calculations involving the value π
we have to use an approximation for the value.
For a rough approximation we can use 3.
Better approximations are 3.14 or 22 .
7
We can also use the π button on a calculator.
Most questions will tell you what approximations to use.
When a calculation has lots of steps we write π as a symbol
throughout and evaluate it at the end, if necessary.
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The circumference of a circle
For any circle,
circumference
π=
diameter
or,
C
π=
d
We can rearrange this to make an formula to find the
circumference of a circle given its diameter.
C = πd
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The circumference of a circle
Use π = 3.14 to find the circumference of this circle.
C = πd
8 cm
= 3.14 × 8
= 25.12 cm
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Finding the circumference given the radius
The diameter of a circle is two times its radius, or
d = 2r
We can substitute this into the formula
C = πd
to give us a formula to find the circumference of a circle
given its radius.
C = 2πr
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The circumference of a circle
Use π = 3.14 to find the circumference of the following circles:
4 cm
C = πd
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C = 2πr
= 3.14 × 4
= 2 × 3.14 × 9
= 12.56 cm
= 56.52 m
C = πd
23 mm
9m
58 cm
C = 2πr
= 3.14 × 23
= 2 × 3.14 × 58
= 72.22 mm
= 364.24 cm
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Finding the radius given the circumference
Use π = 3.14 to find the radius of this circle.
12 cm
C = 2πr
How can we rearrange
this to make r the subject
of the formula?
C
r= ?
2π
12
=
2 × 3.14
= 1.91 cm (to 2 d.p.)
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Find the perimeter of this shape
Use π = 3.14 to find perimeter of this shape.
The perimeter of this shape is
made up of the circumference
of a circle of diameter 13 cm
and two lines of length 6 cm.
6 cm
13 cm
Perimeter = 3.14 × 13 + 6 + 6
= 52.82 cm
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Circumference problem
The diameter of a bicycle wheel is 50 cm. How many
complete rotations does it make over a distance of 1 km?
Using C = πd and π = 3.14,
The circumference of the wheel
= 3.14 × 50
= 157 cm
1 km = 100 000 cm
50 cm
The number of complete rotations
= 100 000 ÷ 157
= 636
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Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
S8.3 Surface area
S8.4 Volume
S8.5 Circumference of a circle
S8.6 Area of a circle
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Area of a circle
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Formula for the area of a circle
We can find the area of a circle using the formula
Area of a circle = π × r × r
radius
or
Area of a circle = πr2
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The circumference of a circle
Use π = 3.14 to find the area of this circle.
4 cm
A = πr2
= 3.14 × 4 × 4
= 50.24 cm2
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Finding the area given the diameter
The radius of a circle is half of its radius, or
d
r=
2
We can substitute this into the formula
A = πr2
to give us a formula to find the area of a circle given its
diameter.
πd2
A=
4
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The area of a circle
Use π = 3.14 to find the area of the following circles:
2 cm
A = πr2
= 3.14 ×
A = πr2
22
10 m
= 12.56 cm2
A = πr2
23 mm
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= 3.14 × 52
= 78.5 m2
78 cm
A = πr2
= 3.14 × 232
= 3.14 × 392
= 1661.06 mm2
= 4775.94 cm2
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Find the area of this shape
Use π = 3.14 to find area of this shape.
The area of this shape is made up
of the area of a circle of diameter
13 cm and the area of a rectangle
of width 6 cm and length 13 cm.
6 cm
13 cm
Area of circle = 3.14 × 6.52
= 132.665 cm2
Area of rectangle = 6 × 13
= 78 cm2
Total area = 132.665 + 78
= 210.665 cm2
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Area of a sector
What is the area of this sector?
72°
5 cm
72°
× π × 52
Area of the sector =
360°
1
=
× π × 52
5
=π×5
= 15.7 cm2 (to 1 d.p.)
We can use this method to find the area of any sector.
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Area problem
Find the shaded area to 2 decimal places.
Area of the square = 12 × 12
= 144 cm2
1
Area of sector =
× π × 122
4
= 36π
Shaded area = 144 – 36π
12 cm
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= 30.9 cm2 (to 1 d.p.)
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