© boardworks ltd 2006 1 of 84 ks3 mathematics s8 perimeter, area and volume

© Boardworks Ltd 2006 1 of 84

KS3 Mathematics

S8 Perimeter, area and volume


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Contents


S8.1 Perimeter

S8.6 Area of a circle

S8.2 Area

S8.5 Circumference of a circle

S8.3 Surface area

S8.4 Volume


Put these shapes in order


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


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)


Perimeter


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.


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.


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


P = 5 + 4 + 4 + 5 + 4 + 4

Perimeter


Remember, the dashes indicate the sides that are the same length.

5 cm

4 cm

= 26 cm


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



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



Equal perimeters

Which shape has a different perimeter from the first shape?

A B C

A B C

A B C

B

A

A


Contents


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S8.2 Area

S8.1 Perimeter



S8.3 Surface area

S8.4 Volume


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


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


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


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


We can use a formula to find the area of a right-angled triangle:


base, b

height, h

Area of a triangle = 12

× base × height

= 12

bh



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


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


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


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?


C D

BA


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.




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



Area of an irregular shape on a pegboard


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


Area of a triangle


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


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


Area of a parallelogram



Area of a parallelogram = base × perpendicular height

base


The area of any parallelogram can be found using the formula:


Area of a parallelogram = bh



What is the area of this parallelogram?


12 cm

7 cm

= 7 × 12

= 84 cm2

8 cm

We can ignore this length


Area of a trapezium


Area of a trapezium

The area of any trapezium can be found using the formula:

Area of a trapezium = (sum of parallel sides) × height12


Area of a trapezium = (a + b)h12


a

b


Area of a trapezium

9 m

6 m

14 m


= (6 + 14) × 912

= × 20 × 912

= 90 m2

What is the area of this trapezium?


Area of a trapezium

What is the area of this trapezium?


= (8 + 3) × 1212

= × 11 × 1212

= 66 m2

8 m

3 m

12 m


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.


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?


Area formulae of 2-D shapes

You should know the following formulae:

b

hArea of a triangle = bh

12



b

h

a

h

b


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.


Contents


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S8.3 Surface area

S8.1 Perimeter


S8.2 Area


S8.4 Volume


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.

Surface area of a cuboid




The front and the back of the cuboid have the same area.





The left hand side and the right hand side of the cuboid have the same area.




Can you work out the surface area of this cuboid?


7 cm

8 cm5 cm

The area of the top = 8 × 5

= 40 cm2

The area of the front = 7 × 5

= 35 cm2

The area of the side = 7 × 8

= 56 cm2



So the total surface area =


7 cm

8 cm5 cm

2 × 40 cm2

+ 2 × 35 cm2

+ 2 × 56 cm2

Top and bottom

Front and back

Left and right side

= 80 + 70 + 112 = 262 cm2


We can find the formula for the surface area of a cuboid as follows.

Surface area of a cuboid =

Formula for the surface area of a cuboid

h

lw

2 × lw Top and bottom

+ 2 × hw Front and back

+ 2 × lh Left and right side

= 2lw + 2hw + 2lh


How can we find the surface area of a cube of length x?

Surface area of a cube

x

All six faces of a cube have the same area.

The area of each face is x × x = x2.

Therefore:

Surface area of a cube = 6x2


This cuboid is made from alternate purple and green centimetre cubes.

Chequered cuboid problem

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?

47 cm2


What is the surface area of this L-shaped prism?

Surface area of a prism

6 cm

5 cm

3 cm

4 cm

3 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 six rectangles that make up the surface of the shape.

Total surface area

= 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15= 110 cm2


5 cm

6 cm

3 cm6 cm

3 cm3 cm

3 cm

It can be helpful to use the net of a 3-D shape to calculate its surface area.

Using nets to find surface area

Here is the net of a 3 cm by 5 cm by 6 cm cuboid.

Write down the area of each face.

15 cm2 15 cm2

18 cm2

30 cm2 30 cm2

18 cm2

Then add the areas together to find the surface area.

Surface Area = 126 cm2


Here is the net of a regular tetrahedron.

Using nets to find surface area

What is its surface area?

6 cm

5.2 cm

Area of each face = ½bh

= ½ × 6 × 5.2

= 15.6 cm2

Surface area = 4 × 15.6

= 62.4 cm2


Contents


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S8.4 Volume

S8.1 Perimeter


S8.2 Area


S8.3 Surface area


The following cuboid is made out of interlocking cubes.

Making cuboids

How many cubes does it contain?


We can work this out by dividing the cuboid into layers.

Making cuboids

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.


The amount of space that a three-dimensional object takes up is called its volume.

Making cuboids

We can use mm3, cm3, m3 or km3.

The 3 tells us that there are three dimensions, length, width and height.

Volume is measured in cubic units.

Liquid volume or capacity is measured in ml, l, pints or gallons.


Volume of a cuboid

We can find the volume of a cuboid by multiplying the area of the base by the height.

Volume of a cuboid

= length × width × height

= lwh

height, h

length, lwidth, w

The area of the base

= length × width

So:


Volume of a cuboid

What is the volume of this cuboid?

Volume of cuboid

= length × width × height

= 13 × 8 × 5

= 520 cm3

5 cm

8 cm 13 cm


Volume and displacement


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 cm31 ml of water has a volume of 1 cm3

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.


What is the volume of this L-shaped prism?

Volume of a prism made from cuboids

6 cm

5 cm

3 cm

4 cm

3 cmWe can think of the shape as two cuboids joined together.

Volume of the green cuboid

= 6 × 3 × 3 = 54 cm3

Volume of the blue cuboid

= 3 × 2 × 2 = 12 cm3

Total volume

= 54 + 12 = 66 cm3


Remember, a prism is a 3-D shape with the same cross-section throughout its length.

Volume of a prism

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

3 cm


What is the volume of this triangular prism?

Volume of a prism

5 cm

4 cm

7.2 cm

Area of cross-section = ½ × 5 × 4 = 10 cm2

Volume of prism = 10 × 7.2 = 72 cm3


Volume of a prism

Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 =

Volume of prism = 5 × 72 = 360 m3

3 m

4 m

12 m

7 m

5 m

72 m2

What is the volume of this prism?


Contents


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S8.1 Perimeter


S8.2 Area

S8.3 Surface area

S8.4 Volume


Circle circumference and diameter


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 = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 (to 200 decimal places)!


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 .227

We can also use the π button on a calculator.

Most questions will tell you which approximation to use.

When a calculation has lots of steps we write π as a symbol throughout and evaluate it at the end, if necessary.


The circumference of a circle

For any circle:

π =circumference

diameter

or:

We can rearrange this to make an formula to find the circumference of a circle given its diameter.

C = πdC = πd

π =Cd



Use π = 3.14 to find the circumference of this circle.

C = πd8 cm

= 3.14 × 8

= 25.12 cm


Finding the circumference given the radius

The diameter of a circle is two times its radius, or

C = 2πrC = 2πr

d = 2rd = 2r

We can substitute this into the formula

C = πdC = πd

to give us a formula to find the circumference of a circle given its radius.



Use π = 3.14 to find the circumference of the following circles:

C = πd4 cm

= 3.14 × 4

= 12.56 cm

C = 2πr9 m

= 2 × 3.14 × 9

= 56.52 m

C = πd

23 mm = 3.14 × 23

= 72.22 mm

C = 2πr58 cm

= 2 × 3.14 × 58

= 364.24 cm


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Finding the radius given the circumference

Use π = 3.14 to find the radius of this circle.

C = 2πrC = 2πr12 cm

How can we rearrange this to make r the subject of the formula?

r =C

2π

12

2 × 3.14≈

= 1.91 cm (to 2 d.p.)


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 cm13 cm

Perimeter = 3.14 × 13 + 6 + 6

= 52.82 cm


Circumference problem

The diameter of a bicycle wheel is 50 cm. How many complete rotations does it make over a distance of 1 km?

50 cm

The circumference of the wheel

= 3.14 × 50

Using C = πd and π = 3.14,

= 157 cm

The number of complete rotations

= 100 000 ÷ 157

≈ 636

1 km = 100 000 cm


Contents


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AS8.6 Area of a circle

S8.1 Perimeter

S8.2 Area


S8.3 Surface area

S8.4 Volume


Area of a circle


Formula for the area of a circle

We can find the area of a circle using the formula.

radius

Area of a circle = πr2Area of a circle = πr2

Area of a circle = π × r × r

or



Use π = 3.14 to find the area of this circle.

A = πr24 cm

≈ 3.14 × 4 × 4

= 50.24 cm2


Finding the area given the diameter

The radius of a circle is half of its diameter, or

We can substitute this into the formula

A = πr2A = πr2

to give us a formula to find the area of a circle given its diameter.

r = d2

A = πd2

4


The area of a circle

Use π = 3.14 to find the area of the following circles:

A = πr22 cm

= 3.14 × 22

= 12.56 cm2

A = πr2

10 m= 3.14 × 52

= 78.5 m2

A = πr2

23 mm = 3.14 × 232

= 1661.06 mm2

A = πr2

78 cm= 3.14 × 392

= 4775.94 cm2


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 cm13 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


Area of a sector

What is the area of this sector?

72°5 cm

Area of the sector =72°

360°× π × 52

1

5= × π × 52

= π × 5

= 15.7 cm2 (to 1 d.p.)

We can use this method to find the area of any sector.


Area problem

Find the shaded area to 2 decimal places.

Area of the square = 12 × 12

1

4Area of sector = × π × 122

= 36π

12 cm

= 144 cm2

Shaded area = 144 – 36π

= 30.96 cm2 (to 2 d.p.)

© boardworks ltd 2006 1 of 84 ks3 mathematics s8 perimeter, area and volume

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