© boardworks ltd 2008 1 of 84 maths age 11-14 s8 perimeter, area and volume
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© Boardworks Ltd 2008 1 of 84
Maths Age 11-14
S8 Perimeter, area and volume
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A
A
Contents
S8 Perimeter, area and volume
S8.1 Perimeter
S8.2 Area
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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?
1 cm
3
3
2
11
2
Perimeter = 3 + 3 + 2 + 1 + 1 + 2
= 12 cm
Starting point
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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 + 2wor
= 2(l + w)
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Perimeter
What is the perimeter of this shape?
b cm
a cm
9 cm
5 cm
12 cm 4 cm
The lengths of two of the sides are not given so we have to work them out before we can find the perimeter.
Let’s call the lengths a and b.
Sometimes we are not given the lengths of all the sides. We have to work them out using the information we are given.
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Perimeter
9 cm
5 cm
b cm
12 cm
a cm
4 cm
a = 12 – 5 cm
= 7 cm
7 cm
b = 9 – 4 cm
= 5 cm
5 cm
P = 9 + 5 + 4 + 7 + 5 + 12
= 42 cm
Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given.
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Calculate the lengths of the missing sides to find the perimeter.
P = 5 + 2 + 1.5 + 6 + 4 + 2 + 10 + 2 + 4 + 6 + 1.5 + 2
Perimeter
5 cm
2 cm
6 cm
4 cm4 cm
2 cm2 cm
p
q r
s
t
u
p = 2 cm
q = r = 1.5 cm
s = 6 cm
t = 2 cm
u =10 cm
= 46 cm
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P = 5 + 4 + 4 + 5 + 4 + 4
Perimeter
What is the perimeter of this shape?
Remember, the dashes indicate the sides that are the same length.
5 cm
4 cm
= 26 cm
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Perimeter
Perimeter = 4.5 + 2 + 1 + 2 + 1 + 2 + 4.5
Start by finding the lengths of all the sides.
5 m
2 m
2 m
2 m
4 m
4.5 m 4.5 m
1 m1 m
= 17 m
What is the perimeter of this shape?
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Before we can find the perimeter we must convert all the lengths to the same units.
Perimeter
3 m
2.4 m
1.9 m
256 cm
In this example, we can either use metres or centimetres.
Using centimetres,
300 cm
240 cm
190 cm
P = 256 + 190 + 240 + 300
= 986 cm
What is the perimeter of this shape?
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Equal perimeters
Which shape has a different perimeter from the first shape?
A B C
A B C
A B C
B
A
A
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Contents
S8 Perimeter, area and volume
A
A
S8.2 Area
S8.1 Perimeter
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The area of a shape is a measure of how much surface the shape takes up.
Area
For example, which of these rugs covers a larger surface?
Rug A
Rug B
Rug C
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Area of a rectangle
Area is measured in square units.
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
Area of a rectangle
= length × width
= lw
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Area of a rectangle
What is the area of this rectangle?
8 cm
4 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?
8 cm
4 cm
What is the shape of the shaded part?
What is the area of this right-angled triangle?
Area of the triangle = × 8 × 4 = 12
4 × 4 = 16 cm2
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We can use a formula to find the area of a right-angled triangle:
Area of a right-angled triangle
base, b
height, h
Area of a triangle = 12
× base × height
= 12
bh
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Area of a right-angled triangle
Calculate the area of this right-angled triangle.
6 cm
8 cm
10 cm
To work out the area of this triangle we only need the length of the base and the height.
We can ignore the third length opposite the right angle.
Area = 12
× base × height
= × 8 × 612
= 24 cm2
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Area of shapes made from rectangles
How can we find the area of this shape?
7 m
10 m
8 m
5 m
15 m
15 m
We can think of this shape as being made up of two rectangles.
Either like this …
… or like this.
Label the rectangles A and B.
A
B 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?
We can think of this shape as being made up of one rectangle with another rectangle cut out of it.
7 cm
8 cm
3 cm
4 cm Label the rectangles A and B.
A
B
Area A = 7 × 8 = 56 cm2
Area B = 3 × 4 = 12 cm2
Total area = 56 – 12 = 44 cm2
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ED
C
B
A
Area of an irregular shapes on a pegboard
We can divide the shape into right-angled triangles and a square.
Area A = ½ × 2 × 3 = 3 units2
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 = 10.5 units2
How can we find the area of this irregular quadrilateral constructed on a pegboard?
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C D
BA
Area of an irregular shapes on a pegboard
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.
How can we find the area of this irregular quadrilateral constructed on a pegboard?
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Area of an irregular shapes on a pegboard
Area A = ½ × 2 × 3 = 3 units2
A B
C D
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
= 10.5 units2
How can we find the area of this irregular quadrilateral constructed on a pegboard?
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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?
8 cm
4 cm
Drawing a line here might help.
What is the area of this triangle?
Area of the triangle = × 8 × 4 = 12
4 × 4 = 16 cm2
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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:
Area of a triangle = × base × perpendicular height12
base
perpendicular height
Or using letter symbols:
Area of a triangle = bh12
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Area of a triangle
What is the area of this triangle?
Area of a triangle = bh12
7 cm
6 cm
= 12
× 7 × 6
= 21 cm2
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Area of a parallelogram
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Area of a parallelogram
Area of a parallelogram = base × perpendicular height
base
perpendicular height
The area of any parallelogram can be found using the formula:
Or using letter symbols:
Area of a parallelogram = bh
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Area of a parallelogram
What is the area of this parallelogram?
Area of a parallelogram = bh
12 cm
7 cm
= 7 × 12
= 84 cm2
8 cm
We can ignore this length
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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:
Area of a trapezium = (sum of parallel sides) × height12
Or using letter symbols:
Area of a trapezium = (a + b)h12
perpendicular height
a
b
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Area of a trapezium
9 m
6 m
14 m
Area of a trapezium = (a + b)h12
= (6 + 14) × 912
= × 20 × 912
= 90 m2
What is the area of this trapezium?
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Area of a trapezium
What is the area of this trapezium?
Area of a trapezium = (a + b)h12
= (8 + 3) × 1212
= × 11 × 1212
= 66 m2
8 m
3 m
12 m
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Area problems
7 cm
10 cm
What is the area of the yellow square?
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
3 cm
This diagram shows a yellow square inside a blue square.
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Area problems
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
3 cm
This diagram shows a yellow square inside a blue square.
What is the area of the yellow square?
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Area formulae of 2-D shapes
You should know the following formulae:
b
hArea of a triangle = bh
12
Area of a parallelogram = bh
Area of a trapezium = (a + b)h12
b
h
a
h
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
1.24 m
518 mm
Let’s write all the lengths in cm.
518 mm = 51.8 cm
1.24 m = 124 cm
Area of the trapezium = ½(76 + 124) × 51.8
= ½ × 200 × 51.8
= 5180 cm2
Don’t forget to put the units at the end.