Not for distribution

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level

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.}

Example:

Find the volume of this fish tank, giving your answer in litres.

Example:

This is a net of a cuboid. If one square has an area of 1 cm2_,

what is the volume of the cuboid?

50 cm V= 50 x 40 x 30 = 60 000 cm3 V = 60 litres V = 4 x 3 x 2 = 24 cm3 40 cm 30 cm

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.

These two cuboids have the same volume. Find the value of x.

Answer

Volume of first cuboid = 36 cm3_.

So volume of second cuboid = 6x = 36 cm3_.

So x = 6 cm.

The volume of the Sun can hold overa million E

arths. 4 cm 3 cm 3 cm 3 cm 2 cm x

Top Tip!

You should know these cube roots: 1 = 1 8 = 2 27 = 3 64 = 4 125 = 5 1000 = 10 3 3 3 3 3 3

Volume of a cuboid

level

5 Sample mental

test question

Sample National

Test question

Surface area of a cuboid

level

6 Volume

level

6 SHAPE, SPACE AND MEASURES

• Volume is the amount of space inside a 3-D shape. • The common units for volume are: mm3_{, cm}3_{or m}3_.

Example:

Find the volume of this cuboid.

length 5 cm 3 cm 12 cm width height

• Volume = length x width x height

V = lx w x h V = lwh

V = lwh

= 12 x 3 x 5 = 180 cm3_.

Top Tip!

Substitute numbers into a formula before trying to work anything out.

• There are 6 faces on a cuboid, with opposite faces having the same area.

• The surface area is given by A = 2lw + 2lh + 2wh

Example:

Find the surface area of the purple 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 cm2_.

pot Check

1 What is the volume and surface

area of this cuboid?

4 cm 3 cm 2 cm

l w

h

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Sample National

Test question

Construct a triangle that has the following properties:

• Total length of three sides is 12 cm.

• Only two of the sides are equal length.

• All sides are whole numbers of centimetres.

Answer

Angle bisector

• The angle bisector is the line that passes at the same distance from

two intersecting lines.

Example:

Construct the angle bisector of ABC.

B A C B A C level

7 Constructing an angle of 60°

Example:

Draw an angle of 60° at the point B. 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°.

level

7

B

5 cm 5 cm

pot Check

1 Draw this triangle accurately. level

7

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 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.

About 4000 years ag

o, the Babylonians tracked the path ofthe Sun a

cross the sky and realised it to

ok a year(about 360 days) to complete o

ne circuit.This led them to divide the c

ircle into 360°.

Constructing triangles

level

6

• When constructing triangles it is very important that you measure lines and

angles accurately.

Example:

Construct this triangle accurately.

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.

Perpendicular bisector

• 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.

Example:

Construct the perpendicular bisector of AB. 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.

Top Tip!

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.

Constructions

SHAPE, SPACE AND MEASURES

4 cm 7 cm 8 cm A B A B level

7

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Loci

• Most loci problems are set in a real-life context.

Example:

A radio transmitter is to be built so that it is the same distance from two 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.

level

8

Towtown Seeton Radville Transmitter could be positioned anywhere on this red line Scale: 1 cm represents 10 km

Top Tip!

Make sure your construction arcs are shown and that the required locus is clearly marked.

Sample National

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.

level

8

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.

T T T T T T T T

The locusof the Ear

th aroundthe Sun is notcircular bu

t elliptical.

Paths

level

7

• A locus (singular of loci) is the path moved by a point according to a rule.

Example:

Draw the locus of all the points that are

a exactly 2 cm from A 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.

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 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.

Loci

SHAPE, SPACE AND MEASURES

A B a b A B a b A B

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levels

5-6

levels

5-6

Volume

1 a What is the volume of this cuboid?

cm3 b What is the surface area?

cm2 1 cm 5 cm 3 cm 1 mark 1 mark

2 A cuboid has a volume of 36 cm3_.

Its length is 6 cm and its width is 3 cm. What is the height of the cuboid?

cm 1 mark

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.

2 marks

4 The volume of a cube is 64 cm3_.

What is the length of each edge of the cube?

cm 1 mark

5 The surface area of this cuboid is 184 cm2_.

Work out the length of the cuboid.

cm

4 cm

5 cm

1 mark

6 This is a net of a cuboid.

What is the volume of the cuboid?

2 m

3 m

8 A tank has the following measurements.

How many litres of water can it hold?

2 m

50 cm 80 cm

2 marks

9 These two cuboids have the same volume. What is the value of x?

8 cm 2 cm 3 cm x 3 cm 4 cm

7 Here are four cuboids.

Rearrange the cuboids in the order of their volume, with the smallest first.

4 cm 4 cm 18 cm 5 cm 4 cm 12 cm 12 cm 5 cm 3 cm 5 cm 25 cm 5 cm A B C D 1 mark litres

SHAPE, SPACE AND MEASURES

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levels

6-7

levels

6-7

1 Construct an angle of 60° at the point A on the line AB.

2 marks

B A

2 Construct the perpendicular bisector of the line AB.

2 marks

3 Construct the angle bisector of the angle ABC.

2 marks

B A

A

B C

4 Draw this triangle accurately.

2 marks

4 cm 7 cm

6 cm

2 marks

7 cm

8 cm 40°

2 marks

8 cm 35° 50°

SHAPE, SPACE AND MEASURES

Constructions

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1 In each of these squares shade the region described.

a All points that are nearer to P than to Q. b All points that are nearer to S than to Q.

levels

7-8

levels

7-8

Loci

P S Q R P S Q R a b 1 mark 1 mark

2 ABCD are squares of side 3 cm. Match the given loci to the diagrams.

i 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.

Loci a

A D B C

Loci b

A D B C

Loci c

A D B C

Loci d

A D B C

3 Construct the locus of the point that is the same distance from the lines AB and AC.

B 4 marks 2 marks 2 marks A B

a Does the radar station at B pick up an aircraft flying directly over A?

b Show all the points where aircraft are picked up by both radar stations.

5 The diagram shows a garden with a garden shed.

Each grid square represents 50 cm.

1 mark

Shed

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.

SHAPE, SPACE AND MEASURES

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

B

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MATHS WORKBOOK 5–8

Pages 66–67 Volume

1 a 15 cm3 _{b 46 cm}2 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 cm}3_{, A = 288 cm}3_, _{B = 625 cm}3 8 800 l (1 mark for 800 000 cm3_{or 0.8 m}3₎ 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)

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) 6 (1 mark for 1 correct side and 1 angle, 1 mark for all correct)

Shape, space and

measures answers

Shape, space and

measures answers

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MATHS WORKBOOK 5–8

Pages 70–71 Loci

1 a b

2 i b ii c iii a iv d

3 (1 mark for arcs)

4 a No b Shown half scale

5

Shape, space and

measures answers

Shape, space and

measures answers

P S Q R P S Q R B A C A B Shed