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04 NCM7 2nd ed SB TXT.fm Page 104 Saturday, June 7, 2008 8:13 PM

SPACE AND GEOMETRY

We live in a three-dimensional (3-D) world. Everywhere around us we see three-dimensional shapes, or solids—cars, buildings, books, trees, ﬂowers and people. Solids are three-dimensional because they have length, width and thickness.

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04 NCM7 2nd ed SB TXT.fm Page 105 Saturday, June 7, 2008 4:24 PM

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04 NCM7 2nd ed SB TXT.fm Page 106 Saturday, June 7, 2008 4:24 PM

Start up Worksheet 4-01

1 a Name six two-dimensional (ﬂat) shapes that can be found in the picture below. b Write a description of each shape you found. Discuss this with other students.

Brainstarters 4

Worksheet 4-02 A page of shapes

Skillsheet 4-01 Naming shapes

TLF

L 1059

Shape maker: replicator

Geometry 4-01 The vocabulary of geometry

2 Draw each of the following shapes. a a pentagon b an equilateral triangle d a trapezium e a hexagon

c a semi-circle f a parallelogram

3 How many sides has: a a rectangle? d a hexagon?

c a triangle? f a pentagon?

b an octagon? e a quadrilateral?

4 What type of lines point in the same direction and never meet? 5 a Name each of the following solids.

C

B

A

F

G

E

D

H

I

b Name everyday uses for each of the solids in part a. 6 Which of the solids from Question 5 has: a ﬁve faces? b six vertices (corners)? d no ﬂat surfaces? e all faces triangular? g all faces square? h all faces rectangular? j no rectangular faces (Some have more than one answer!)

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c curved surfaces? f two triangular faces? i one square face?

04 NCM7 2nd ed SB TXT.fm Page 107 Saturday, June 7, 2008 4:24 PM

4-01 Convex and non-convex solids Here are examples of some solid shapes you should know.

Worksheet 4-02 A page of shapes

Skillsheet 4-01 Naming shapes

Triangular pyramid

Rectangular prism

Sphere

Cylinder

Square pyramid

Pentagonal prism

These solids are convex.

These solids are non-convex.

Convex solids have faces that all point, curve or bulge outwards, while non-convex solids have some faces that point, curve or cave inwards, having ‘dents’ or ‘holes’. A simple test to determine whether a solid is convex or non-convex is to draw straight lines between any two faces on the solid. If every line you can draw lies completely inside the solid, then the solid is convex.

If all or part of any line you can draw lies outside the solid, then the solid is non-convex.

CHAPTER 4 SOLIDS

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Exercise 4-01 1 Test whether each of these solids is convex or non-convex. a b c

Worksheet 4-03

d

e

f

g

h

i

2 a Copy the following nets, or print out Worksheet 4-03. b Cut out each net along the solid lines. c Fold each net along the dotted lines to make a solid, but do not paste the edges together. d Write the name of each solid on one of the light green faces. e State if each solid is convex or non-convex. f Unfold each net and paste it in your workbook on its darker green face.

Nets of solids 1

i

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ii

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iii

iv

v

vi

vii

viii

CHAPTER 4 SOLIDS

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ix

x

3 Bring to class any other examples of solid shapes you can ﬁnd—the more the better!

4-02 Nets of solids The nets in the following exercise have tabs for pasting or taping the edges of each solid together. For best results, copy them onto cardboard ﬁrst.

Exercise 4-02

4 cm

4 cm

m

4c

4c m

4 cm

NEW CENTURY MATHS 7

4 cm

4 cm

4 cm

110

m

4 cm

4c

4c m

1 Here is the net for a triangular prism. Using the measurements in the net, copy it to make a model of a triangular prism.

4 cm

4 cm

04 NCM7 2nd ed SB TXT.fm Page 111 Saturday, June 7, 2008 4:24 PM

Worksheet 4-04 Nets of solids 2

tab

tab

2 a Copy the nets below, or print out Worksheet 4-04. b Cut out each net along its boundary. c Fold each net along the dotted lines to make a solid. i

tab

tab

Triangular prism

tab

ii

tab

tab

Oblique triangular prism

tab

tab

tab CHAPTER 4 SOLIDS

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iii

tab

tab

tab

tab

Cube

tab

tab

tab

iv

b

ta

tab

tab

Pentagonal pyramid

tab

tab

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tab

v

tab

Oblique square pyramid

tab

tab tab

vi

tab

ta b

Oblique triangular pyramid

CHAPTER 4 SOLIDS

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vii

tab tab

Triangular pyramid

tab

viii

tab

tab

tab

tab

tab

tab

Trapezoidal prism

tab

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3 Match the correct shape name to each net (a to h). (Some have more than one net.) triangular prism cube rectangular prism square pyramid cylinder trapezoidal prism a

b

c

d

e

f

g

h

CHAPTER 4 SOLIDS

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4 Which of the following cannot be folded to make a cube? Select A, B, C or D. A B C D

4-03 Polyhedra TLF

L 1061

Shape maker: complex objects 1

TLF

face

Solids have flat faces and curved faces. This cylinder has two ﬂat faces and one curved face. The ﬂat faces are circles, and the curved face is a rectangle (when ﬂattened).

curved face

L 1062

Shape maker: complex objects 2

hexagon

This hexagonal prism has eight faces. Two of the faces are hexagons and the other six faces are rectangles.

face

rectangle

A solid whose faces are all ﬂat is called a polyhedron. The plural of ‘polyhedron’ is ‘polyhedra’ or ‘polyhedrons’. A hexagonal prism is a polyhedron. A cylinder, however, is not because it has a curved face. Just as polygons such as pentagons, hexagons and octagons are named according to the number of sides they have, polyhedra are named according to the number of faces they have. For example, a tetrahedron has four faces (tetra = 4), and a hexahedron has six faces (hexa = 6).

Tetrahedron (4 faces)

Hexahedron (6 faces)

This table shows the names of some polyhedra. Polyhedron

Number of faces

Polyhedron

Number of faces

Tetrahedron

4

Nonahedron

9

Pentahedron

5

Decahedron

10

Hexahedron

6

Undecahedron

11

Heptahedron

7

Dodecahedron

12

Octahedron

8

Icosahedron

20

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Exercise 4-03 1 What is a more common name for a tetrahedron? 2 Find out what a tetrapak and the computer game Tetris are. How are they related to the meaning of ‘tetra’? 3 Which of the following solids are polyhedra? a a rectangular prism b a triangular prism d a square pyramid e a cone

c a sphere f a donut shape

4 Two solids have been combined to make each of these composite ﬁgures. For each one: i name the two solids that have been combined ii state whether the composite ﬁgure is a polyhedron iii state whether the composite ﬁgure is convex. a b c

d

e

f

5 Copy the ﬁgures from Question 4 parts a, b and f and colour one pair of parallel ﬂat faces. 6 Copy and complete this table. Solid

Number of faces

Shapes of faces

Number of identical faces

Number of pairs of parallel ﬂat faces

Cube Cylinder Square pyramid Triangular prism Rectangular prism Rectangular pyramid Cone Triangular pyramid

7 What is the smallest number of faces a polyhedron can have? CHAPTER 4 SOLIDS

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8 How many faces has: a a tetrahedron? c a hexahedron?

b an octahedron? d a pentahedron?

9 Use the table of polyhedra names on page 116 to name these polyhedra. a

b

c

d

e

f

10 Complete this statement, selecting A, B, C or D. The name of this polyhedron is: A hexahedron B decahedron C pentahedron D octahedron 11 Why can a cube, a rectangular prism and a pentagonal pyramid all be called hexahedrons? 12 Name any solid whose faces are all identical.

Working mathematically

Reasoning and reﬂecting

Stacking and packaging 1 Look around your classroom. Why are bricks like rectangular prisms? Why is the room a rectangular prism? What shape are most books? Why do you think this is? 2 Imagine you are in a supermarket. What solid shapes are used for packaging? Why do you think different shapes are used? How are items stacked on the shelves? What part does packaging play in this? 3 Find out what solid shape bees use to make a beehive. Why do they use it? 4 Find other three-dimensional structures in your environment. What solid shapes have been used? Why might this be so?

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4-04 Special solids A cross-section of a solid is a ‘slice’ of the solid, cut across it, parallel to its end faces, rather than along it. These diagrams show cross-sections of a rectangular prism, a sphere and a square pyramid. cross-section cross-section

cross-section

Rectangular prism

Sphere

Square pyramid

Note that the cross-sections of a rectangular prism are congruent (identical) rectangles, the cross-sections of a sphere are circles (but of different sizes), and the cross-sections of a square pyramid are squares (again, of different sizes).

Prisms and pyramids Prisms and pyramids are special types of polyhedra. A prism has the same (uniform) cross-section along its length. Each cross-section is a polygon. Either of the end faces is called the prism’s base. Prisms take their names from their base. The prism shown below has trapezium-shaped cross-sections, identical and parallel to its base. This is a trapezoidal prism.

base

cross-section

Trapezoidal prism

A prism is either right or oblique, as shown in the diagrams of triangular prisms below.

not right angles

right angles

Right prism

Oblique prism

A right prism stands upright, while an oblique prism stands at an angle or slant. ‘Oblique’ means ‘slanting’ or ‘at an angle’, while ‘right’ means ‘at right angles’. A pyramid has a pointed top called the apex. The face opposite the apex is a polygon and is called the pyramid’s base. A pyramid’s cross-sections have the same shape as the base but are not the same size. The pyramid on the right has a triangular base, so it is called a triangular pyramid.

apex

cross-section

base

Triangular pyramid

CHAPTER 4 SOLIDS

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A pyramid is also either right or oblique, as shown in the diagrams of square pyramids on the right.

Right pyramid

Oblique pyramid

Cylinders, cones and spheres Cylinders, cones and spheres are not polyhedra because they have curved faces. Although it has a uniform cross-section, a cylinder is not a prism because its cross-sections are circles, which are not polygons. Although it has an apex, a cone is not a pyramid because its base is a circle, which is not a polygon. Like prisms and pyramids, cylinders and cones can be either right or oblique, as shown below.

Right cylinder

Oblique cylinder

Oblique cone

Right cone

A sphere is a perfectly circular solid, the shape of a ball. All of the points on a sphere’s surface are exactly the same distance from the centre of the sphere. In this diagram of a sphere, O is the centre and P and Q are points on the surface of the sphere. The distance OP is the same as the distance OQ.

P Q

O

Example 1 a b c d e f

Draw a cross-section of the solid shown. Is the cross-section of this solid uniform (the same)? Is this solid a prism or a pyramid? Is this solid right or oblique? What shape is the solid’s base? What shape are the side faces? g What is the full name of this solid?

Solution a

b Yes, all cross-sections of this solid are identical pentagons. c A prism, because the cross-section is uniform and is a polygon.

d Oblique because it is slanted. f The side faces are parallelograms.

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NEW CENTURY MATHS 7

e The base is a pentagon. g This solid is an oblique pentagonal prism.

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Exercise 4-04 1 Draw a cross-section of each of these solids. a b c

f

k

g

h

l

Ex 1

d

e

i

j

m

2 Which solids in Question 1 are: i polyhedra? iii pyramids?

n

ii prisms? iv neither prisms nor pyramids?

3 Which solid shown is a prism? Select A, B, C or D. a b c

4 Describe in your own words: a a prism

d

b a pyramid.

5 For each of these prisms, state: i the shape of its cross-section ii its name iii whether it is a right prism or an oblique prism. a b c

d

e

f

CHAPTER 4 SOLIDS

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6 Using each of these shapes as a base, draw a prism and shade its base. a square b isosceles triangle c trapezium d hexagon 7 Name these pyramids and classify them as right pyramids or oblique pyramids. a b c

d

e

8 What shape are the side faces of: a any right prism? b any pyramid?

f

c any oblique prism?

9 Draw a pentagonal pyramid and shade its base. 10 Explain why a cylinder is not a prism and why a cone is not a pyramid.

Just for the record

Rubik’s cube Rubik’s Cube is a mechanical puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernö Rubik. It is said to be the world’s best-selling toy, with over 300 000 000 Rubik’s Cubes sold around the world. In Rubik’s Cube, 26 individual little cubes or cubies make up the big cube. Each layer of nine cubies can twist and the layers can overlap. Typically, the faces of the cube are covered by nine stickers in six solid colours; there is one colour for each side of the cube. When the puzzle is solved, each face of the cube is the same colour. Every two years, the Rubik’s Cube world championship is held to ﬁnd the person who can solve the puzzle fastest. The world record of 9.86 seconds was set by Thibaut Jacquinot of France in May 2007. The current Australian champion is Jasmine Lee with a personal best of 20.36 seconds. Anssi Vanhala of Finland won the feet-only challenge in 2007, aligning the colours of the 3 × 3 cube in 49.33 seconds.

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4-05 Classifying solids Throughout this chapter, we have seen that solids can be classiﬁed in different ways. A solid may belong to one or more of the categories shown below. Category

Meaning

Polyhedron

Any solid whose faces are all ﬂat

Prism

A polyhedron with a uniform polygonal cross-section

Right prism

An upright prism

Oblique prism

A slanted prism

Pyramid

A polyhedron with a polygonal base and an apex

Right pyramid

An upright pyramid

Oblique pyramid

A slanted pyramid

Example

CHAPTER 4 SOLIDS

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Category

Meaning

Example

Right cylinder

An upright cylinder

Oblique cylinder

A slanted cylinder

Right cone

An upright solid with a circular base and an apex

Oblique cone

A slanted cone

Sphere

A ball shape that is completely round, with all points on its surface the same distance from the centre

Exercise 4-05 1 For each of these solids: i write its name a b

e

124

f

NEW CENTURY MATHS 7

ii state whether it is right, oblique or neither. c d

g

h

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2 Answer these questions for the following shapes. a Which are convex? b Which are prisms? c Which are oblique? d Which are pyramids? A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

CHAPTER 4 SOLIDS

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Y

Z

CC

DD

AA

BB

EE

FF

3 Which solids have been used to form the solid shown on the right? Select A, B, C or D. A cylinder and rectangular prism B semicircle and trapezoidal prism C cylinder and trapezoidal prism D sphere and trapezium 4 A slice of a cone is cut vertically as shown in the diagram on the right. What shape is the slice? Select A, B, C or D from the diagrams below. A B

C

D

4-06 Euler’s rule Worksheet 4-05 Euler’s rule

Leonhard Euler (1707–83) was a famous Swiss mathematician who discovered an interesting rule about polyhedra. He developed a formula relating the number of faces, vertices and edges of a convex polyhedron. edge An edge is a line of the solid, where two faces meet. vertex A vertex is a corner of the solid, where edges meet. (plural: The prism in the diagram has six faces, eight vertices) face vertices and 12 edges.

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Exercise 4-06 1 Count the number of faces (F ), vertices (V) and edges (E) of the polyhedra drawn below and of any others you have collected. Copy and complete the table below.

Polyhedron

Number of faces (F )

Number of vertices (V )

6

8

Cube

Number of edges (E )

F +V

Rectangular prism Triangular prism Triangular pyramid Square pyramid

2 Examine your results in the table from Question 1, where F is the number of faces, V is the number of vertices and E is the number of edges. Select A, B, C or D to complete Euler’s rule. Euler’s rule: F+V=E+ ? A2

B 18

C4

D 12

3 A piece has been sliced off each solid shown below. Count the number of faces (F ), the number of vertices (V ) and the number of edges (E ) of each remaining solid. Does Euler’s rule still hold true for them? a

b

c

d

4 Identify each of the following solids. a I have six vertices and nine edges. b I have six faces and they are all the same shape. c I have the same number of vertices as faces, but my faces are not all the same. d I have six edges. e I have 12 edges and eight vertices, and my faces are not all the same.

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!

Euler’s rule for convex polyhedra is: Faces + vertices = edges + 2 F +V = E + 2

Mental skills 4

Maths without calculators

Estimating answers A quick way of estimating an answer is to round each number in the calculation. 1 Examine these examples. a 631 + 280 + 51 + 43 + 96 ≈ 600 + 300 + 50 + 40 + 100 = (600 + 300 + 100) + (50 + 40) = 1000 + 90 = 1090 (Actual answer = 1101) b 55 + 132 − 34 + 17 − 78 ≈ 60 + 130 − 30 + 20 − 80 = (60 + 20 − 80) + (130 − 30) = 0 + 100 = 100 (Actual answer = 92) c 67 × 13 ≈ 70 × 12 = 840

(Actual answer = 871)

d 78 × 7 ≈ 80 × 7 e 929 ÷ 5 ≈ 1000 ÷ 5 = 200

(Actual answer = 185.8)

f 510 ÷ 24 ≈ 500 ÷ 20 ≈ 50 ÷ 2 = 25

(Actual answer = 21.25)

2 Now estimate these answers. a 27 + 11 + 87 + 142 + 64

b 55 + 34 − 22 − 46 + 136

c 684 + 903

d 35 + 81 + 110 + 22 + 7

e 517 − 96

f

g 766 − 353

h 367 × 2

210 − 38 − 71 + 151 − 49

83 × 81

j

984 × 16

k 828 ÷ 3

l

507 ÷ 7

i

4-07 Edges of a solid Two edges of a solid can be related in three different ways. They can be intersecting, parallel or skew.

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In the prisms shown at the right: • AD and CD are intersecting edges. Intersecting edges meet at a vertex. Edges AD and CD meet at the vertex D.

C

B A

D

F E

• AB and HG are parallel edges, written AB II HG. Parallel edges point in the same direction and never meet.

G H

B A

G H B

• BF and EH are skew edges. Skew edges point in different directions but never meet because they are on different planes. Both edges could not lie on the same ﬂat surface (such as a sheet of paper). Skew means ‘not on the same plane’.

F E

H

Exercise 4-07 1 In this diagram of a rectangular prism, name all the edges that: a intersect at vertex G b are parallel to edge CG c are skew to edge AD.

B A

C D F

E

G H

2 Which edge is skew to DC? Select A, B, C or D. A GH B BC C EF D AG

B A

C D

F E

G H

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3 For each of the following solids, name a pair of: i intersecting edges ii parallel edges iii skew edges. a b c A B D H

N

J A

E

D G

C

E

M

O

C F

K

L

F

B

G

H J

d

e

K

A

f

S T

M

L

R

V

P

Q

U

O

N

Q

R

h

U P

A

I

B

J

V Y

D

W

B

E

H

D

C

C

H F

I

X

N

C

i

A

S T

B

D

P

W

g

E Q

G

E

F

G

4 Copy and complete the statements next to each of these solids. Choose from ‘parallel’, ‘intersecting’ or ‘skew’ to ﬁll the blanks. S a ST and UY are T

R

VW and WX are

U

W

TX and RV are

X

V

WY and TX are

Y

b

A

AE and AC are BE and CD are DE and AB are

C B

D

BD and AB are

E

c

G

GK and HL are

H I

J

MN and KL are GJ and LM are

K N

130

HI and IM are

L M

NEW CENTURY MATHS 7

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Working mathematically

Communicating and reﬂecting

Perpendicular, parallel and skew lines These pictures show examples of perpendicular, parallel and skew lines in the world around us.

Skew lines

Parallel lines

Perpendicular lines Parallel lines

Parallel lines Parallel lines

Perpendicular lines Skew lines

Make a collage of pictures from magazines and newspapers and highlight the different types of lines.

4-08 The Platonic solids A regular polygon is a ﬂat shape with equal sides. For example, a regular hexagon has six equal sides. A regular polyhedron or Platonic solid is a convex polyhedron with identical faces. Every face is the same regular polygon. For example, a cube is a regular polyhedron because every face is a square. Because a cube has six identical faces, it is also called a regular hexahedron. There are only ﬁve Platonic solids, as listed in the table and drawn on the next page. Platonic solid

Number of equal faces

Regular tetrahedron

tetra = 4

Regular hexahedron

hexa = 6

Regular octahedron

octa = 8

Regular dodecahedron Regular icosahedron

dodeca = 12 icosa = 20

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Hexahedron Tetrahedron (equilateral (cube) triangular pyramid)

Octahedron

Dodecahedron

Icosahedron

The Platonic solids were discovered by the Greeks in the 5th century BC. However, ancient artefacts displaying pictures of these solids have been found in Europe, Egypt, Africa and South America. It was believed that these shapes had mystical properties. They are named after the Greek philosopher Plato (427–327 BC) who claimed that these solids were ‘cosmic bodies’ representing the elements. • The tetrahedron stood for Fire. • The hexahedron stood for Earth. • The octahedron stood for Air. • The icosahedron represented Water. • The dodecahedron represented the ether or the Universe.

Exercise 4-08

NEW CENTURY MATHS 7

m

5 cm

m 5c

m 5c

2 A regular octahedron is shown on the right. a Draw and cut out the net of a regular octahedron with side lengths of 5 cm. Fold and paste to make the octahedron.

132

m 5c

5 cm m 5c

1 A regular tetrahedron is shown on the right. a Draw and cut out the net of a regular tetrahedron with side lengths of 5 cm. For best results, draw it on to cardboard. Fold along the dotted lines and paste the tabs to make the tetrahedron. b How many faces has the regular tetrahedron? c What is the more common name for a tetrahedron? d What type of triangle is each face? e How many vertices has this tetrahedron? f How many edges has this tetrahedron? g Does this tetrahedron have any: i parallel edges? ii skew edges? iii intersecting edges?

5c

Nets of Platonic solids

You can print out Worksheet 4-06 for the nets used in this exercise.

5c m

Worksheet 4-06

5 cm

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5c

m

5 cm

m

5c

b How many faces has this regular octahedron? c What is the shape of each face? d A regular octahedron is a composite solid. What two solids can be combined to make it? e How many edges has this regular octahedron? f Does this octahedron have any: i parallel edges? ii skew edges? iii intersecting edges?

4c

m

4c

m

3 A regular icosahedron is shown on the right. a Draw and cut out the net of a regular icosahedron with side lengths of 4 cm. Then make the icosahedron. b How many faces has this regular icosahedron? c What is the shape of each face? d How many edges does it have? e How many vertices does it have?

4 cm

f Is Euler’s rule true for this icosahedron? g Does this icosahedron have any: i parallel edges?

ii skew edges?

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4 A regular hexahedron is shown on the right. a Draw and cut out this net of a regular hexahedron with side lengths of 5 cm. Then make the hexahedron. b How many faces has the regular hexahedron? c What is the shape of each face? d What is the more common name for a regular hexahedron?

5 cm 5 cm 5 cm

5 A regular dodecahedron is shown on the right. a Draw and cut out this net of a regular dodecahedron with side lengths of 3 cm, then make it. b How many faces has the regular dodecahedron? c What is the shape of each face? d How many edges does it have? e How many vertices does it have? f Is Euler’s rule true for this dodecahedron? g Does this dodecahedron have any: i parallel edges? ii skew edges?

3c

3 cm

3 cm

3 cm

m

134

NEW CENTURY MATHS 7

m

3c

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6 Which of the following is a Platonic solid? Select A, B, C or D. A

B

C D

7 Copy and complete this table of Platonic solids. Platonic solid

Description

Shape of each face

Number of faces

Two square pyramids joined at bases Similar to a soccer ball pattern

Regular pentagon

Icosahedron

20 4 Square

8 Which Platonic solid: a has 12 faces? d has 12 vertices?

b is a pyramid? e is a prism?

Working mathematically

c has 12 edges? f has pentagonal faces?

Communicating and reasoning

Special solids Use the library and/or the Internet to research at least one of the following topics. Present your work as a wall chart, in a project book or as a PowerPoint presentation. 1 There are 13 Archimedean solids. Find out their names and how they are related to the Platonic solids. Find pictures of them. 2 There are four Kepler-Poinsot solids. Find out their names and what is special about them. Find pictures of them. 3 What are Johnson solids? How many are there? 4 Find out about how the Pythagoreans, Theaetetus and Plato discovered the Platonic solids. 5 Find out what the mathematician Euclid and the astronomer Kepler discovered about the Platonic solids. 6 Research the lives of Plato, Euler or the Pythagoreans.

CHAPTER 4 SOLIDS

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4-09 Drawing and building solids Solids can be difﬁcult to draw. Sometimes it is easier to use square dot paper or isometric dot paper to draw them. The rectangular prisms shown below are drawn on square dot paper, so called because the dots sit on a grid of squares.

face horizontal edge

The rectangular prisms shown below are drawn on isometric dot paper, with the dots arranged in a triangular pattern. Horizontal edges are drawn at a slant on this paper. A 2-D (ﬂat) diagram of a 3-D ﬁgure (solid) looks more natural when drawn on isometric paper.

hor iz edgontal e

vertical edge

When drawing on square paper, one of the solid’s faces is shown at the front. When drawing on isometric paper, one of the solid’s vertical edges is shown at the front.

Exercise 4-09 Worksheet Appendix 8 Square dot paper

Worksheet Appendix 9

For this exercise you will need square dot paper, isometric dot paper and construction cubes. Both types of dot paper can be printed out using these links. You can also draw shapes directly into a geometry program, using a square grid or an isometric grid. 1 Copy the rectangular prisms shown above: i on square dot paper ii on isometric dot paper.

Isometric dot paper

2 Use cubes to make each of the following solids, then draw them on isometric dot paper. a b c d

e

136

f

NEW CENTURY MATHS 7

g

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3 Draw this shape on isometric dot paper.

4 Make three different solids using ﬁve cubes. Then draw them on isometric dot paper. 5 Each of the following is the start of a drawing of a rectangular prism. Copy each drawing on to isometric dot paper and complete it. a

b

c

d

e

6 Imagine and then draw what would remain of this large block if the smaller block was taken away from it.

The use of cubes should help you complete the following questions. 7 Imagine and then draw what you would see of the following shape if you were looking from A, then from B, and then from C. C

A

B

8 Copy each of these drawings of solids on to isometric dot paper exactly as shown. a b c

4 cubes 5 cubes

6 cubes

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d

e

f

7 cubes 10 cubes 18 cubes

9 Here are four different shapes, called A, B, C and D. A

B

C

D

Which of the following shapes are the same as: a A? b B? c C? i

iv

viii

ii

v

d D? iii

vi

ix

vii

x

10 On isometric dot paper draw the different solids that can be made if four cubes are joined together. Two have already been drawn below for you. How many different solids are there?

138

NEW CENTURY MATHS 7

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11 Which of the given solids is the same as the solid on the left? Select A, B, C, D or E. A

D

B

C

E

4-10 Different views of solids Example 2 For the solid on the right, assume that there are no hidden cubes. Draw: a the front view b the left view c the top view.

nt

fro

lef

t

Solution Imagine you walk around the solid. This is what you would see: a viewed from the front b viewed from the left

c viewed from the top.

CHAPTER 4 SOLIDS

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Example 3 For the solid shown on the right, draw: a the front view b the top view.

front

Solution a

b

Exercise 4-10 Ex 2

1 For each of these solids, draw the views requested. a b

fro

nt

i ii iii

nt

t

top view right view front view

i ii iii

c

fro

left view back view top view

d

fro

nt

i ii iii

140

lef

ht

rig

back view top view right view

NEW CENTURY MATHS 7

lef

t

t igh

r

i ii iii

right view front view top view

nt

fro

04 NCM7 2nd ed SB TXT.fm Page 141 Saturday, June 7, 2008 4:24 PM

e

f

fro

i ii iii

lef

ht

nt

nt

t

rig

front view left view top view

i ii iii

fro

back view right view top view

2 For each of these solids, draw the views requested. a b

front

front

i ii

Ex 3

front view top view

i ii iii d

fro

nt

c

front view right view top view

front

i ii iii

left view top view front view

e

i ii

top view front view

f

front

i ii iii iv

front view left view right view top view

front

i ii

top view front view

CHAPTER 4 SOLIDS

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3 Chuan ﬂew over the top of this building. Which view would he see when directly overhead? Select A, B, C or D.

A

B

C

D

4 What would Diana see if she were looking at this solid from point E? Select A, B, C or D.

E

A

B

C

D

5 Draw each of these solids from the top and front views. a b

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c

d

6 In each set of diagrams below, three views of a solid shape are shown, along with the number of cubes needed to make it. Make each of these solids, then draw them on isometric paper. a 5 cubes b 6 cubes

top

front

side

top

front

side

c 5 cubes

top

front

side

front

side

d 5 cubes

top

Using technology

More spreadsheet formulas Remember that a spreadsheet is like a calculator. We use special symbols to do calculations and a spreadsheet formula always begins with an equals (=) sign. • =A1/A2 means divide the value in cell A1 by the value in cell A2. • =A5^3 means cube (or raise to the power of three) the value in cell A5. • =sqrt(C1) means calculate the square root of the value in cell C1 (that is, C1 ). • =max(B1:B8) means ﬁnd the largest value when comparing the values in cells B1 to B8. • =min(B1:B8) means ﬁnd the smallest value when comparing the values in cells B1 to B8.

CHAPTER 4 SOLIDS

143

Worksheet Appendix 9 Isometric dot paper

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1 Enter the following numbers into cells as shown below, where s represents the value in cell B1, t is the value in cell B2, u is the value in cell B3, and so on.

Note: To enter fractions (as in cell B4), you may need to Format Cells, then choose Fraction and select the number of digits. 2 Enter the following formulas into the given cells. a b c d e f g h

C1, C2, C3, C4, C5, C6, C7, C8,

v (means enter the formula into cell C1 as shown above) 50 − w u3 s×t+u×v (s + t) × w w4 + s 3 − t 2 the maximum value from the set of cells B1 to B5 the minimum value from the set of cells B1 to B5

s+u+v i C9, -------------------6 (When you have obtained the answer, right click on the cell, choose Format Cells and convert the answer to a fraction.) j C10, t + w Hint: Careful with use of brackets! k C11, 3 ÷ (u × v) (Format Cell as in i above to convert to a mixed numeral.) l C12, (t − s) ÷ (w − t) (Format cell as in i above to convert to a fraction.) 3 Now choose different values and enter them into cells B1 to B5. Consider the new answers obtained in column C, for the formulas entered from Question 2. Note: Change the formatting to/from fractions where appropriate. 4 a Now that you have mastered using the basic symbols for doing calculations in a spreadsheet, create your own values and formulas in your workbook. Try to create ten new calculations using a maximum of ﬁve different values. b Now enter your new formulas into the spreadsheet and check if the answers are what you expected! Make any changes to the formulas if necessary. c Swap the values and formulas you have created in part a with a friend. Enter the new values in column D and the new formulas (using appropriate symbols) in column E of your spreadsheet.

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Power plus 1 Draw each of the following. a a non-regular hexahedron

b a regular octahedron

c a pentahedron

2 How many diagonals has: a a cube?

b a square pyramid?

c a triangular prism?

3 Euler’s rule also works in two dimensions. These shapes are called networks, made up of regions (spaces), vertices (corners) and edges (lines or curves joining two vertices). Note: Outside the network also counts as one region. The left network below has 3 regions, 5 vertices and 6 edges. The right network has 3 regions, 4 vertices and 5 edges. vertex region

edg

e

Count the number of regions (R), vertices (V) and edges (E) on each of the following networks, then write Euler’s rule for networks. a b c d

e

f

i

j

g

k

h

l

4 A soccer ball is not a perfect sphere but is actually a polyhedron with 32 faces. a Look at the pattern on a soccer ball to ﬁnd the shapes of its faces, and count how many faces of each shape there are. b A soccer ball is not a regular polyhedron (Platonic solid) but it is a semi-regular polyhedron. Find out what semi-regular polyhedra are. There are 13 of them. 5 The words in this list are related to solids. Research the meanings of ﬁve of them. axis altitude ellipsoid elliptic cone frustum lateral faces oblique cylinder parallelepiped torus truncated prism

CHAPTER 4 SOLIDS

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Chapter 4 review Worksheet 4-07

Language of maths apex convex dodecahedron hexahedron oblique pyramid Platonic solid pyramid right pyramid sphere

Solids crossword

base cross-section edge intersecting edges octahedron polygon regular polygon skew edges tetrahedron

cone cube Euler’s rule icosahedron parallel edges polyhedron/polyhedra regular polyhedron slant uniform

congruent cylinder face oblique prism plane prism right prism solid vertex/vertices

1 Find the ﬁve Platonic solids in the list above. 2 What is the difference between the base of a prism and the base of a pyramid? 3 What is a ‘polyhedron’? Give an example of a polyhedron and an example that is not a polyhedron. 4 What type of polyhedron has a uniform cross-section? 5 What is the difference between a right prism and an oblique prism? 6 What do the Greek preﬁxes ‘hexa’ and ‘poly’ mean? 7 What is the difference between a convex polygon and a non-convex polygon?

Topic overview • How useful do you think this chapter will be to you in the future? • Can you name any jobs which use some of the concepts covered in this chapter? • Did you have any problems with any sections of this chapter? Discuss any problems with a friend or your teacher. • Nets • Models

Views of solids Drawing and building solids

Solids with curved surfaces

SOLIDS Platonic solids Polyhedra Pyramids Prisms

Euler’s rule Right Face

Right

Vertices

Oblique Parallel

146

Edges

NEW CENTURY MATHS 7

Skew

Intersecting

Oblique

04 NCM7 2nd ed SB TXT.fm Page 147 Saturday, June 7, 2008 4:24 PM

Chapter revision

Topic test 4

1 State whether each of these solids is convex or non-convex. a

Exercise 4-01

b

2 Which shape below is the net of a triangular prism? Select A, B, C or D. A

B

C

Exercise 4-02

D

3 Copy and complete the table below for the three solids shown. A B C

Solid A

Solid B

Exercise 4-03

Solid C

Number of faces Number of congruent faces Shapes of the faces Number of parallel ﬂat faces Shape of a cross-section Are the cross-sections uniform? Is the solid a polyhedron?

4 Draw each of the following solids and shade its base. a a hexagonal prism b a cylinder

Exercise 4-04

c a rectangular prism

5 Draw a cross-section of each of these solids and state whether each one is a prism. a b c

d

e

Exercise 4-04

f

CHAPTER 4 SOLIDS

147

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Exercise 4-04

6 Name each of the following solids and state whether each one is right or oblique. a b c

d

e

f

Exercise 4-01

7 Which solids from Questions 3, 5 and 6 are non-convex?

Exercise 4-05

8 Match each shape with its correct property. Shape

Exercise 4-06

Exercise 4-07

Property

a octahedron

A another name for a cube

b pyramid

B a polyhedron with triangular faces and a square base

c sphere

C has all side faces meeting at a point called the apex

d octagonal prism

D a Platonic solid with eight equilateral triangles as faces

e regular hexahedron

E has a uniform cross-section but is not a prism

f

F a polyhedron with 10 faces, 16 vertices and 24 edges

oblique prism

g square pyramid

G another name for a triangular pyramid

h cylinder

H a box shape that is slanted

i

right pyramid

I

a solid with no ﬂat faces

j

tetrahedron

J

a pyramid that is upright

9 Draw a triangular prism and count the number of: a faces b edges 10 For each of these solids, name a pair of: i intersecting edges ii parallel edges a b N T

iii skew edges c A

U

W P

R

B

V

O

NEW CENTURY MATHS 7

L

M

S J

148

c vertices

K

E

F

D

C

04 NCM7 2nd ed SB TXT.fm Page 149 Saturday, June 7, 2008 4:24 PM

11 What is a Platonic solid? Give an example of one.

Exercise 4-08

12 This solid is a regular octahedron.

Exercise 4-08

a b c d

Which two solids can be combined to make this ﬁgure? How is this octahedron regular? How many faces, vertices and edges has this octahedron? Describe Euler’s rule for convex polyhedra and show that it is true for this octahedron.

Exercise 4-06

13 Copy and complete these rectangular prisms on dot paper. a b

Exercise 4-09

14 For each of these solids, sketch the views requested.

Exercise 4-10

a

i ii iii

front view left view top view

i ii iii

top view right view front view

i ii iii

right view front view top view

ht

fro

rig

nt

b

ht

fro

rig

nt

c

front

CHAPTER 4 SOLIDS

149

View more...
SPACE AND GEOMETRY

We live in a three-dimensional (3-D) world. Everywhere around us we see three-dimensional shapes, or solids—cars, buildings, books, trees, ﬂowers and people. Solids are three-dimensional because they have length, width and thickness.

4

1234

48901234

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04 NCM7 2nd ed SB TXT.fm Page 105 Saturday, June 7, 2008 4:24 PM

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• describe solids and classify them as convex or 1234567890123456789 5678 • congruent Another word for ‘identical’. 234567890123 45678901234 0123456789 non-convex • convex solid A solid that points or curves 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 8 7890123456789012 3456789012345678 • make models of solids outwards, so that a line joining any two of its faces 789012345678 901234567890123456789012345678901234567890123456789012345678 lies completely inside it. 567890123456789012345678901234567890123456789012345 • describe solids in terms of their geometrical 0123456789 1234567890123456789 properties 56789012345678901234567890 78901234 5678 234567890123 45678901234• cross-section A ‘slice’ of a solid, cut across it 0123456789 • identify any pairs of parallel ﬂat 7890123456789012 surfaces of a solid 345678901 rather than along it. 4567890156789012345678901234567890123456 456789012 9012345 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04 NCM7 2nd ed SB TXT.fm Page 106 Saturday, June 7, 2008 4:24 PM

Start up Worksheet 4-01

1 a Name six two-dimensional (ﬂat) shapes that can be found in the picture below. b Write a description of each shape you found. Discuss this with other students.

Brainstarters 4

Worksheet 4-02 A page of shapes

Skillsheet 4-01 Naming shapes

TLF

L 1059

Shape maker: replicator

Geometry 4-01 The vocabulary of geometry

2 Draw each of the following shapes. a a pentagon b an equilateral triangle d a trapezium e a hexagon

c a semi-circle f a parallelogram

3 How many sides has: a a rectangle? d a hexagon?

c a triangle? f a pentagon?

b an octagon? e a quadrilateral?

4 What type of lines point in the same direction and never meet? 5 a Name each of the following solids.

C

B

A

F

G

E

D

H

I

b Name everyday uses for each of the solids in part a. 6 Which of the solids from Question 5 has: a ﬁve faces? b six vertices (corners)? d no ﬂat surfaces? e all faces triangular? g all faces square? h all faces rectangular? j no rectangular faces (Some have more than one answer!)

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NEW CENTURY MATHS 7

c curved surfaces? f two triangular faces? i one square face?

04 NCM7 2nd ed SB TXT.fm Page 107 Saturday, June 7, 2008 4:24 PM

4-01 Convex and non-convex solids Here are examples of some solid shapes you should know.

Worksheet 4-02 A page of shapes

Skillsheet 4-01 Naming shapes

Triangular pyramid

Rectangular prism

Sphere

Cylinder

Square pyramid

Pentagonal prism

These solids are convex.

These solids are non-convex.

Convex solids have faces that all point, curve or bulge outwards, while non-convex solids have some faces that point, curve or cave inwards, having ‘dents’ or ‘holes’. A simple test to determine whether a solid is convex or non-convex is to draw straight lines between any two faces on the solid. If every line you can draw lies completely inside the solid, then the solid is convex.

If all or part of any line you can draw lies outside the solid, then the solid is non-convex.

CHAPTER 4 SOLIDS

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Exercise 4-01 1 Test whether each of these solids is convex or non-convex. a b c

Worksheet 4-03

d

e

f

g

h

i

2 a Copy the following nets, or print out Worksheet 4-03. b Cut out each net along the solid lines. c Fold each net along the dotted lines to make a solid, but do not paste the edges together. d Write the name of each solid on one of the light green faces. e State if each solid is convex or non-convex. f Unfold each net and paste it in your workbook on its darker green face.

Nets of solids 1

i

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ii

04 NCM7 2nd ed SB TXT.fm Page 109 Saturday, June 7, 2008 4:24 PM

iii

iv

v

vi

vii

viii

CHAPTER 4 SOLIDS

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ix

x

3 Bring to class any other examples of solid shapes you can ﬁnd—the more the better!

4-02 Nets of solids The nets in the following exercise have tabs for pasting or taping the edges of each solid together. For best results, copy them onto cardboard ﬁrst.

Exercise 4-02

4 cm

4 cm

m

4c

4c m

4 cm

NEW CENTURY MATHS 7

4 cm

4 cm

4 cm

110

m

4 cm

4c

4c m

1 Here is the net for a triangular prism. Using the measurements in the net, copy it to make a model of a triangular prism.

4 cm

4 cm

04 NCM7 2nd ed SB TXT.fm Page 111 Saturday, June 7, 2008 4:24 PM

Worksheet 4-04 Nets of solids 2

tab

tab

2 a Copy the nets below, or print out Worksheet 4-04. b Cut out each net along its boundary. c Fold each net along the dotted lines to make a solid. i

tab

tab

Triangular prism

tab

ii

tab

tab

Oblique triangular prism

tab

tab

tab CHAPTER 4 SOLIDS

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iii

tab

tab

tab

tab

Cube

tab

tab

tab

iv

b

ta

tab

tab

Pentagonal pyramid

tab

tab

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tab

v

tab

Oblique square pyramid

tab

tab tab

vi

tab

ta b

Oblique triangular pyramid

CHAPTER 4 SOLIDS

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vii

tab tab

Triangular pyramid

tab

viii

tab

tab

tab

tab

tab

tab

Trapezoidal prism

tab

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3 Match the correct shape name to each net (a to h). (Some have more than one net.) triangular prism cube rectangular prism square pyramid cylinder trapezoidal prism a

b

c

d

e

f

g

h

CHAPTER 4 SOLIDS

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4 Which of the following cannot be folded to make a cube? Select A, B, C or D. A B C D

4-03 Polyhedra TLF

L 1061

Shape maker: complex objects 1

TLF

face

Solids have flat faces and curved faces. This cylinder has two ﬂat faces and one curved face. The ﬂat faces are circles, and the curved face is a rectangle (when ﬂattened).

curved face

L 1062

Shape maker: complex objects 2

hexagon

This hexagonal prism has eight faces. Two of the faces are hexagons and the other six faces are rectangles.

face

rectangle

A solid whose faces are all ﬂat is called a polyhedron. The plural of ‘polyhedron’ is ‘polyhedra’ or ‘polyhedrons’. A hexagonal prism is a polyhedron. A cylinder, however, is not because it has a curved face. Just as polygons such as pentagons, hexagons and octagons are named according to the number of sides they have, polyhedra are named according to the number of faces they have. For example, a tetrahedron has four faces (tetra = 4), and a hexahedron has six faces (hexa = 6).

Tetrahedron (4 faces)

Hexahedron (6 faces)

This table shows the names of some polyhedra. Polyhedron

Number of faces

Polyhedron

Number of faces

Tetrahedron

4

Nonahedron

9

Pentahedron

5

Decahedron

10

Hexahedron

6

Undecahedron

11

Heptahedron

7

Dodecahedron

12

Octahedron

8

Icosahedron

20

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Exercise 4-03 1 What is a more common name for a tetrahedron? 2 Find out what a tetrapak and the computer game Tetris are. How are they related to the meaning of ‘tetra’? 3 Which of the following solids are polyhedra? a a rectangular prism b a triangular prism d a square pyramid e a cone

c a sphere f a donut shape

4 Two solids have been combined to make each of these composite ﬁgures. For each one: i name the two solids that have been combined ii state whether the composite ﬁgure is a polyhedron iii state whether the composite ﬁgure is convex. a b c

d

e

f

5 Copy the ﬁgures from Question 4 parts a, b and f and colour one pair of parallel ﬂat faces. 6 Copy and complete this table. Solid

Number of faces

Shapes of faces

Number of identical faces

Number of pairs of parallel ﬂat faces

Cube Cylinder Square pyramid Triangular prism Rectangular prism Rectangular pyramid Cone Triangular pyramid

7 What is the smallest number of faces a polyhedron can have? CHAPTER 4 SOLIDS

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8 How many faces has: a a tetrahedron? c a hexahedron?

b an octahedron? d a pentahedron?

9 Use the table of polyhedra names on page 116 to name these polyhedra. a

b

c

d

e

f

10 Complete this statement, selecting A, B, C or D. The name of this polyhedron is: A hexahedron B decahedron C pentahedron D octahedron 11 Why can a cube, a rectangular prism and a pentagonal pyramid all be called hexahedrons? 12 Name any solid whose faces are all identical.

Working mathematically

Reasoning and reﬂecting

Stacking and packaging 1 Look around your classroom. Why are bricks like rectangular prisms? Why is the room a rectangular prism? What shape are most books? Why do you think this is? 2 Imagine you are in a supermarket. What solid shapes are used for packaging? Why do you think different shapes are used? How are items stacked on the shelves? What part does packaging play in this? 3 Find out what solid shape bees use to make a beehive. Why do they use it? 4 Find other three-dimensional structures in your environment. What solid shapes have been used? Why might this be so?

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4-04 Special solids A cross-section of a solid is a ‘slice’ of the solid, cut across it, parallel to its end faces, rather than along it. These diagrams show cross-sections of a rectangular prism, a sphere and a square pyramid. cross-section cross-section

cross-section

Rectangular prism

Sphere

Square pyramid

Note that the cross-sections of a rectangular prism are congruent (identical) rectangles, the cross-sections of a sphere are circles (but of different sizes), and the cross-sections of a square pyramid are squares (again, of different sizes).

Prisms and pyramids Prisms and pyramids are special types of polyhedra. A prism has the same (uniform) cross-section along its length. Each cross-section is a polygon. Either of the end faces is called the prism’s base. Prisms take their names from their base. The prism shown below has trapezium-shaped cross-sections, identical and parallel to its base. This is a trapezoidal prism.

base

cross-section

Trapezoidal prism

A prism is either right or oblique, as shown in the diagrams of triangular prisms below.

not right angles

right angles

Right prism

Oblique prism

A right prism stands upright, while an oblique prism stands at an angle or slant. ‘Oblique’ means ‘slanting’ or ‘at an angle’, while ‘right’ means ‘at right angles’. A pyramid has a pointed top called the apex. The face opposite the apex is a polygon and is called the pyramid’s base. A pyramid’s cross-sections have the same shape as the base but are not the same size. The pyramid on the right has a triangular base, so it is called a triangular pyramid.

apex

cross-section

base

Triangular pyramid

CHAPTER 4 SOLIDS

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A pyramid is also either right or oblique, as shown in the diagrams of square pyramids on the right.

Right pyramid

Oblique pyramid

Cylinders, cones and spheres Cylinders, cones and spheres are not polyhedra because they have curved faces. Although it has a uniform cross-section, a cylinder is not a prism because its cross-sections are circles, which are not polygons. Although it has an apex, a cone is not a pyramid because its base is a circle, which is not a polygon. Like prisms and pyramids, cylinders and cones can be either right or oblique, as shown below.

Right cylinder

Oblique cylinder

Oblique cone

Right cone

A sphere is a perfectly circular solid, the shape of a ball. All of the points on a sphere’s surface are exactly the same distance from the centre of the sphere. In this diagram of a sphere, O is the centre and P and Q are points on the surface of the sphere. The distance OP is the same as the distance OQ.

P Q

O

Example 1 a b c d e f

Draw a cross-section of the solid shown. Is the cross-section of this solid uniform (the same)? Is this solid a prism or a pyramid? Is this solid right or oblique? What shape is the solid’s base? What shape are the side faces? g What is the full name of this solid?

Solution a

b Yes, all cross-sections of this solid are identical pentagons. c A prism, because the cross-section is uniform and is a polygon.

d Oblique because it is slanted. f The side faces are parallelograms.

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NEW CENTURY MATHS 7

e The base is a pentagon. g This solid is an oblique pentagonal prism.

04 NCM7 2nd ed SB TXT.fm Page 121 Saturday, June 7, 2008 4:24 PM

Exercise 4-04 1 Draw a cross-section of each of these solids. a b c

f

k

g

h

l

Ex 1

d

e

i

j

m

2 Which solids in Question 1 are: i polyhedra? iii pyramids?

n

ii prisms? iv neither prisms nor pyramids?

3 Which solid shown is a prism? Select A, B, C or D. a b c

4 Describe in your own words: a a prism

d

b a pyramid.

5 For each of these prisms, state: i the shape of its cross-section ii its name iii whether it is a right prism or an oblique prism. a b c

d

e

f

CHAPTER 4 SOLIDS

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6 Using each of these shapes as a base, draw a prism and shade its base. a square b isosceles triangle c trapezium d hexagon 7 Name these pyramids and classify them as right pyramids or oblique pyramids. a b c

d

e

8 What shape are the side faces of: a any right prism? b any pyramid?

f

c any oblique prism?

9 Draw a pentagonal pyramid and shade its base. 10 Explain why a cylinder is not a prism and why a cone is not a pyramid.

Just for the record

Rubik’s cube Rubik’s Cube is a mechanical puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernö Rubik. It is said to be the world’s best-selling toy, with over 300 000 000 Rubik’s Cubes sold around the world. In Rubik’s Cube, 26 individual little cubes or cubies make up the big cube. Each layer of nine cubies can twist and the layers can overlap. Typically, the faces of the cube are covered by nine stickers in six solid colours; there is one colour for each side of the cube. When the puzzle is solved, each face of the cube is the same colour. Every two years, the Rubik’s Cube world championship is held to ﬁnd the person who can solve the puzzle fastest. The world record of 9.86 seconds was set by Thibaut Jacquinot of France in May 2007. The current Australian champion is Jasmine Lee with a personal best of 20.36 seconds. Anssi Vanhala of Finland won the feet-only challenge in 2007, aligning the colours of the 3 × 3 cube in 49.33 seconds.

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4-05 Classifying solids Throughout this chapter, we have seen that solids can be classiﬁed in different ways. A solid may belong to one or more of the categories shown below. Category

Meaning

Polyhedron

Any solid whose faces are all ﬂat

Prism

A polyhedron with a uniform polygonal cross-section

Right prism

An upright prism

Oblique prism

A slanted prism

Pyramid

A polyhedron with a polygonal base and an apex

Right pyramid

An upright pyramid

Oblique pyramid

A slanted pyramid

Example

CHAPTER 4 SOLIDS

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Category

Meaning

Example

Right cylinder

An upright cylinder

Oblique cylinder

A slanted cylinder

Right cone

An upright solid with a circular base and an apex

Oblique cone

A slanted cone

Sphere

A ball shape that is completely round, with all points on its surface the same distance from the centre

Exercise 4-05 1 For each of these solids: i write its name a b

e

124

f

NEW CENTURY MATHS 7

ii state whether it is right, oblique or neither. c d

g

h

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2 Answer these questions for the following shapes. a Which are convex? b Which are prisms? c Which are oblique? d Which are pyramids? A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

CHAPTER 4 SOLIDS

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Y

Z

CC

DD

AA

BB

EE

FF

3 Which solids have been used to form the solid shown on the right? Select A, B, C or D. A cylinder and rectangular prism B semicircle and trapezoidal prism C cylinder and trapezoidal prism D sphere and trapezium 4 A slice of a cone is cut vertically as shown in the diagram on the right. What shape is the slice? Select A, B, C or D from the diagrams below. A B

C

D

4-06 Euler’s rule Worksheet 4-05 Euler’s rule

Leonhard Euler (1707–83) was a famous Swiss mathematician who discovered an interesting rule about polyhedra. He developed a formula relating the number of faces, vertices and edges of a convex polyhedron. edge An edge is a line of the solid, where two faces meet. vertex A vertex is a corner of the solid, where edges meet. (plural: The prism in the diagram has six faces, eight vertices) face vertices and 12 edges.

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Exercise 4-06 1 Count the number of faces (F ), vertices (V) and edges (E) of the polyhedra drawn below and of any others you have collected. Copy and complete the table below.

Polyhedron

Number of faces (F )

Number of vertices (V )

6

8

Cube

Number of edges (E )

F +V

Rectangular prism Triangular prism Triangular pyramid Square pyramid

2 Examine your results in the table from Question 1, where F is the number of faces, V is the number of vertices and E is the number of edges. Select A, B, C or D to complete Euler’s rule. Euler’s rule: F+V=E+ ? A2

B 18

C4

D 12

3 A piece has been sliced off each solid shown below. Count the number of faces (F ), the number of vertices (V ) and the number of edges (E ) of each remaining solid. Does Euler’s rule still hold true for them? a

b

c

d

4 Identify each of the following solids. a I have six vertices and nine edges. b I have six faces and they are all the same shape. c I have the same number of vertices as faces, but my faces are not all the same. d I have six edges. e I have 12 edges and eight vertices, and my faces are not all the same.

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!

Euler’s rule for convex polyhedra is: Faces + vertices = edges + 2 F +V = E + 2

Mental skills 4

Maths without calculators

Estimating answers A quick way of estimating an answer is to round each number in the calculation. 1 Examine these examples. a 631 + 280 + 51 + 43 + 96 ≈ 600 + 300 + 50 + 40 + 100 = (600 + 300 + 100) + (50 + 40) = 1000 + 90 = 1090 (Actual answer = 1101) b 55 + 132 − 34 + 17 − 78 ≈ 60 + 130 − 30 + 20 − 80 = (60 + 20 − 80) + (130 − 30) = 0 + 100 = 100 (Actual answer = 92) c 67 × 13 ≈ 70 × 12 = 840

(Actual answer = 871)

d 78 × 7 ≈ 80 × 7 e 929 ÷ 5 ≈ 1000 ÷ 5 = 200

(Actual answer = 185.8)

f 510 ÷ 24 ≈ 500 ÷ 20 ≈ 50 ÷ 2 = 25

(Actual answer = 21.25)

2 Now estimate these answers. a 27 + 11 + 87 + 142 + 64

b 55 + 34 − 22 − 46 + 136

c 684 + 903

d 35 + 81 + 110 + 22 + 7

e 517 − 96

f

g 766 − 353

h 367 × 2

210 − 38 − 71 + 151 − 49

83 × 81

j

984 × 16

k 828 ÷ 3

l

507 ÷ 7

i

4-07 Edges of a solid Two edges of a solid can be related in three different ways. They can be intersecting, parallel or skew.

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In the prisms shown at the right: • AD and CD are intersecting edges. Intersecting edges meet at a vertex. Edges AD and CD meet at the vertex D.

C

B A

D

F E

• AB and HG are parallel edges, written AB II HG. Parallel edges point in the same direction and never meet.

G H

B A

G H B

• BF and EH are skew edges. Skew edges point in different directions but never meet because they are on different planes. Both edges could not lie on the same ﬂat surface (such as a sheet of paper). Skew means ‘not on the same plane’.

F E

H

Exercise 4-07 1 In this diagram of a rectangular prism, name all the edges that: a intersect at vertex G b are parallel to edge CG c are skew to edge AD.

B A

C D F

E

G H

2 Which edge is skew to DC? Select A, B, C or D. A GH B BC C EF D AG

B A

C D

F E

G H

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3 For each of the following solids, name a pair of: i intersecting edges ii parallel edges iii skew edges. a b c A B D H

N

J A

E

D G

C

E

M

O

C F

K

L

F

B

G

H J

d

e

K

A

f

S T

M

L

R

V

P

Q

U

O

N

Q

R

h

U P

A

I

B

J

V Y

D

W

B

E

H

D

C

C

H F

I

X

N

C

i

A

S T

B

D

P

W

g

E Q

G

E

F

G

4 Copy and complete the statements next to each of these solids. Choose from ‘parallel’, ‘intersecting’ or ‘skew’ to ﬁll the blanks. S a ST and UY are T

R

VW and WX are

U

W

TX and RV are

X

V

WY and TX are

Y

b

A

AE and AC are BE and CD are DE and AB are

C B

D

BD and AB are

E

c

G

GK and HL are

H I

J

MN and KL are GJ and LM are

K N

130

HI and IM are

L M

NEW CENTURY MATHS 7

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Working mathematically

Communicating and reﬂecting

Perpendicular, parallel and skew lines These pictures show examples of perpendicular, parallel and skew lines in the world around us.

Skew lines

Parallel lines

Perpendicular lines Parallel lines

Parallel lines Parallel lines

Perpendicular lines Skew lines

Make a collage of pictures from magazines and newspapers and highlight the different types of lines.

4-08 The Platonic solids A regular polygon is a ﬂat shape with equal sides. For example, a regular hexagon has six equal sides. A regular polyhedron or Platonic solid is a convex polyhedron with identical faces. Every face is the same regular polygon. For example, a cube is a regular polyhedron because every face is a square. Because a cube has six identical faces, it is also called a regular hexahedron. There are only ﬁve Platonic solids, as listed in the table and drawn on the next page. Platonic solid

Number of equal faces

Regular tetrahedron

tetra = 4

Regular hexahedron

hexa = 6

Regular octahedron

octa = 8

Regular dodecahedron Regular icosahedron

dodeca = 12 icosa = 20

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Hexahedron Tetrahedron (equilateral (cube) triangular pyramid)

Octahedron

Dodecahedron

Icosahedron

The Platonic solids were discovered by the Greeks in the 5th century BC. However, ancient artefacts displaying pictures of these solids have been found in Europe, Egypt, Africa and South America. It was believed that these shapes had mystical properties. They are named after the Greek philosopher Plato (427–327 BC) who claimed that these solids were ‘cosmic bodies’ representing the elements. • The tetrahedron stood for Fire. • The hexahedron stood for Earth. • The octahedron stood for Air. • The icosahedron represented Water. • The dodecahedron represented the ether or the Universe.

Exercise 4-08

NEW CENTURY MATHS 7

m

5 cm

m 5c

m 5c

2 A regular octahedron is shown on the right. a Draw and cut out the net of a regular octahedron with side lengths of 5 cm. Fold and paste to make the octahedron.

132

m 5c

5 cm m 5c

1 A regular tetrahedron is shown on the right. a Draw and cut out the net of a regular tetrahedron with side lengths of 5 cm. For best results, draw it on to cardboard. Fold along the dotted lines and paste the tabs to make the tetrahedron. b How many faces has the regular tetrahedron? c What is the more common name for a tetrahedron? d What type of triangle is each face? e How many vertices has this tetrahedron? f How many edges has this tetrahedron? g Does this tetrahedron have any: i parallel edges? ii skew edges? iii intersecting edges?

5c

Nets of Platonic solids

You can print out Worksheet 4-06 for the nets used in this exercise.

5c m

Worksheet 4-06

5 cm

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5c

m

5 cm

m

5c

b How many faces has this regular octahedron? c What is the shape of each face? d A regular octahedron is a composite solid. What two solids can be combined to make it? e How many edges has this regular octahedron? f Does this octahedron have any: i parallel edges? ii skew edges? iii intersecting edges?

4c

m

4c

m

3 A regular icosahedron is shown on the right. a Draw and cut out the net of a regular icosahedron with side lengths of 4 cm. Then make the icosahedron. b How many faces has this regular icosahedron? c What is the shape of each face? d How many edges does it have? e How many vertices does it have?

4 cm

f Is Euler’s rule true for this icosahedron? g Does this icosahedron have any: i parallel edges?

ii skew edges?

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4 A regular hexahedron is shown on the right. a Draw and cut out this net of a regular hexahedron with side lengths of 5 cm. Then make the hexahedron. b How many faces has the regular hexahedron? c What is the shape of each face? d What is the more common name for a regular hexahedron?

5 cm 5 cm 5 cm

5 A regular dodecahedron is shown on the right. a Draw and cut out this net of a regular dodecahedron with side lengths of 3 cm, then make it. b How many faces has the regular dodecahedron? c What is the shape of each face? d How many edges does it have? e How many vertices does it have? f Is Euler’s rule true for this dodecahedron? g Does this dodecahedron have any: i parallel edges? ii skew edges?

3c

3 cm

3 cm

3 cm

m

134

NEW CENTURY MATHS 7

m

3c

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6 Which of the following is a Platonic solid? Select A, B, C or D. A

B

C D

7 Copy and complete this table of Platonic solids. Platonic solid

Description

Shape of each face

Number of faces

Two square pyramids joined at bases Similar to a soccer ball pattern

Regular pentagon

Icosahedron

20 4 Square

8 Which Platonic solid: a has 12 faces? d has 12 vertices?

b is a pyramid? e is a prism?

Working mathematically

c has 12 edges? f has pentagonal faces?

Communicating and reasoning

Special solids Use the library and/or the Internet to research at least one of the following topics. Present your work as a wall chart, in a project book or as a PowerPoint presentation. 1 There are 13 Archimedean solids. Find out their names and how they are related to the Platonic solids. Find pictures of them. 2 There are four Kepler-Poinsot solids. Find out their names and what is special about them. Find pictures of them. 3 What are Johnson solids? How many are there? 4 Find out about how the Pythagoreans, Theaetetus and Plato discovered the Platonic solids. 5 Find out what the mathematician Euclid and the astronomer Kepler discovered about the Platonic solids. 6 Research the lives of Plato, Euler or the Pythagoreans.

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4-09 Drawing and building solids Solids can be difﬁcult to draw. Sometimes it is easier to use square dot paper or isometric dot paper to draw them. The rectangular prisms shown below are drawn on square dot paper, so called because the dots sit on a grid of squares.

face horizontal edge

The rectangular prisms shown below are drawn on isometric dot paper, with the dots arranged in a triangular pattern. Horizontal edges are drawn at a slant on this paper. A 2-D (ﬂat) diagram of a 3-D ﬁgure (solid) looks more natural when drawn on isometric paper.

hor iz edgontal e

vertical edge

When drawing on square paper, one of the solid’s faces is shown at the front. When drawing on isometric paper, one of the solid’s vertical edges is shown at the front.

Exercise 4-09 Worksheet Appendix 8 Square dot paper

Worksheet Appendix 9

For this exercise you will need square dot paper, isometric dot paper and construction cubes. Both types of dot paper can be printed out using these links. You can also draw shapes directly into a geometry program, using a square grid or an isometric grid. 1 Copy the rectangular prisms shown above: i on square dot paper ii on isometric dot paper.

Isometric dot paper

2 Use cubes to make each of the following solids, then draw them on isometric dot paper. a b c d

e

136

f

NEW CENTURY MATHS 7

g

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3 Draw this shape on isometric dot paper.

4 Make three different solids using ﬁve cubes. Then draw them on isometric dot paper. 5 Each of the following is the start of a drawing of a rectangular prism. Copy each drawing on to isometric dot paper and complete it. a

b

c

d

e

6 Imagine and then draw what would remain of this large block if the smaller block was taken away from it.

The use of cubes should help you complete the following questions. 7 Imagine and then draw what you would see of the following shape if you were looking from A, then from B, and then from C. C

A

B

8 Copy each of these drawings of solids on to isometric dot paper exactly as shown. a b c

4 cubes 5 cubes

6 cubes

CHAPTER 4 SOLIDS

137

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d

e

f

7 cubes 10 cubes 18 cubes

9 Here are four different shapes, called A, B, C and D. A

B

C

D

Which of the following shapes are the same as: a A? b B? c C? i

iv

viii

ii

v

d D? iii

vi

ix

vii

x

10 On isometric dot paper draw the different solids that can be made if four cubes are joined together. Two have already been drawn below for you. How many different solids are there?

138

NEW CENTURY MATHS 7

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11 Which of the given solids is the same as the solid on the left? Select A, B, C, D or E. A

D

B

C

E

4-10 Different views of solids Example 2 For the solid on the right, assume that there are no hidden cubes. Draw: a the front view b the left view c the top view.

nt

fro

lef

t

Solution Imagine you walk around the solid. This is what you would see: a viewed from the front b viewed from the left

c viewed from the top.

CHAPTER 4 SOLIDS

139

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Example 3 For the solid shown on the right, draw: a the front view b the top view.

front

Solution a

b

Exercise 4-10 Ex 2

1 For each of these solids, draw the views requested. a b

fro

nt

i ii iii

nt

t

top view right view front view

i ii iii

c

fro

left view back view top view

d

fro

nt

i ii iii

140

lef

ht

rig

back view top view right view

NEW CENTURY MATHS 7

lef

t

t igh

r

i ii iii

right view front view top view

nt

fro

04 NCM7 2nd ed SB TXT.fm Page 141 Saturday, June 7, 2008 4:24 PM

e

f

fro

i ii iii

lef

ht

nt

nt

t

rig

front view left view top view

i ii iii

fro

back view right view top view

2 For each of these solids, draw the views requested. a b

front

front

i ii

Ex 3

front view top view

i ii iii d

fro

nt

c

front view right view top view

front

i ii iii

left view top view front view

e

i ii

top view front view

f

front

i ii iii iv

front view left view right view top view

front

i ii

top view front view

CHAPTER 4 SOLIDS

141

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3 Chuan ﬂew over the top of this building. Which view would he see when directly overhead? Select A, B, C or D.

A

B

C

D

4 What would Diana see if she were looking at this solid from point E? Select A, B, C or D.

E

A

B

C

D

5 Draw each of these solids from the top and front views. a b

142

NEW CENTURY MATHS 7

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c

d

6 In each set of diagrams below, three views of a solid shape are shown, along with the number of cubes needed to make it. Make each of these solids, then draw them on isometric paper. a 5 cubes b 6 cubes

top

front

side

top

front

side

c 5 cubes

top

front

side

front

side

d 5 cubes

top

Using technology

More spreadsheet formulas Remember that a spreadsheet is like a calculator. We use special symbols to do calculations and a spreadsheet formula always begins with an equals (=) sign. • =A1/A2 means divide the value in cell A1 by the value in cell A2. • =A5^3 means cube (or raise to the power of three) the value in cell A5. • =sqrt(C1) means calculate the square root of the value in cell C1 (that is, C1 ). • =max(B1:B8) means ﬁnd the largest value when comparing the values in cells B1 to B8. • =min(B1:B8) means ﬁnd the smallest value when comparing the values in cells B1 to B8.

CHAPTER 4 SOLIDS

143

Worksheet Appendix 9 Isometric dot paper

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1 Enter the following numbers into cells as shown below, where s represents the value in cell B1, t is the value in cell B2, u is the value in cell B3, and so on.

Note: To enter fractions (as in cell B4), you may need to Format Cells, then choose Fraction and select the number of digits. 2 Enter the following formulas into the given cells. a b c d e f g h

C1, C2, C3, C4, C5, C6, C7, C8,

v (means enter the formula into cell C1 as shown above) 50 − w u3 s×t+u×v (s + t) × w w4 + s 3 − t 2 the maximum value from the set of cells B1 to B5 the minimum value from the set of cells B1 to B5

s+u+v i C9, -------------------6 (When you have obtained the answer, right click on the cell, choose Format Cells and convert the answer to a fraction.) j C10, t + w Hint: Careful with use of brackets! k C11, 3 ÷ (u × v) (Format Cell as in i above to convert to a mixed numeral.) l C12, (t − s) ÷ (w − t) (Format cell as in i above to convert to a fraction.) 3 Now choose different values and enter them into cells B1 to B5. Consider the new answers obtained in column C, for the formulas entered from Question 2. Note: Change the formatting to/from fractions where appropriate. 4 a Now that you have mastered using the basic symbols for doing calculations in a spreadsheet, create your own values and formulas in your workbook. Try to create ten new calculations using a maximum of ﬁve different values. b Now enter your new formulas into the spreadsheet and check if the answers are what you expected! Make any changes to the formulas if necessary. c Swap the values and formulas you have created in part a with a friend. Enter the new values in column D and the new formulas (using appropriate symbols) in column E of your spreadsheet.

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Power plus 1 Draw each of the following. a a non-regular hexahedron

b a regular octahedron

c a pentahedron

2 How many diagonals has: a a cube?

b a square pyramid?

c a triangular prism?

3 Euler’s rule also works in two dimensions. These shapes are called networks, made up of regions (spaces), vertices (corners) and edges (lines or curves joining two vertices). Note: Outside the network also counts as one region. The left network below has 3 regions, 5 vertices and 6 edges. The right network has 3 regions, 4 vertices and 5 edges. vertex region

edg

e

Count the number of regions (R), vertices (V) and edges (E) on each of the following networks, then write Euler’s rule for networks. a b c d

e

f

i

j

g

k

h

l

4 A soccer ball is not a perfect sphere but is actually a polyhedron with 32 faces. a Look at the pattern on a soccer ball to ﬁnd the shapes of its faces, and count how many faces of each shape there are. b A soccer ball is not a regular polyhedron (Platonic solid) but it is a semi-regular polyhedron. Find out what semi-regular polyhedra are. There are 13 of them. 5 The words in this list are related to solids. Research the meanings of ﬁve of them. axis altitude ellipsoid elliptic cone frustum lateral faces oblique cylinder parallelepiped torus truncated prism

CHAPTER 4 SOLIDS

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Chapter 4 review Worksheet 4-07

Language of maths apex convex dodecahedron hexahedron oblique pyramid Platonic solid pyramid right pyramid sphere

Solids crossword

base cross-section edge intersecting edges octahedron polygon regular polygon skew edges tetrahedron

cone cube Euler’s rule icosahedron parallel edges polyhedron/polyhedra regular polyhedron slant uniform

congruent cylinder face oblique prism plane prism right prism solid vertex/vertices

1 Find the ﬁve Platonic solids in the list above. 2 What is the difference between the base of a prism and the base of a pyramid? 3 What is a ‘polyhedron’? Give an example of a polyhedron and an example that is not a polyhedron. 4 What type of polyhedron has a uniform cross-section? 5 What is the difference between a right prism and an oblique prism? 6 What do the Greek preﬁxes ‘hexa’ and ‘poly’ mean? 7 What is the difference between a convex polygon and a non-convex polygon?

Topic overview • How useful do you think this chapter will be to you in the future? • Can you name any jobs which use some of the concepts covered in this chapter? • Did you have any problems with any sections of this chapter? Discuss any problems with a friend or your teacher. • Nets • Models

Views of solids Drawing and building solids

Solids with curved surfaces

SOLIDS Platonic solids Polyhedra Pyramids Prisms

Euler’s rule Right Face

Right

Vertices

Oblique Parallel

146

Edges

NEW CENTURY MATHS 7

Skew

Intersecting

Oblique

04 NCM7 2nd ed SB TXT.fm Page 147 Saturday, June 7, 2008 4:24 PM

Chapter revision

Topic test 4

1 State whether each of these solids is convex or non-convex. a

Exercise 4-01

b

2 Which shape below is the net of a triangular prism? Select A, B, C or D. A

B

C

Exercise 4-02

D

3 Copy and complete the table below for the three solids shown. A B C

Solid A

Solid B

Exercise 4-03

Solid C

Number of faces Number of congruent faces Shapes of the faces Number of parallel ﬂat faces Shape of a cross-section Are the cross-sections uniform? Is the solid a polyhedron?

4 Draw each of the following solids and shade its base. a a hexagonal prism b a cylinder

Exercise 4-04

c a rectangular prism

5 Draw a cross-section of each of these solids and state whether each one is a prism. a b c

d

e

Exercise 4-04

f

CHAPTER 4 SOLIDS

147

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Exercise 4-04

6 Name each of the following solids and state whether each one is right or oblique. a b c

d

e

f

Exercise 4-01

7 Which solids from Questions 3, 5 and 6 are non-convex?

Exercise 4-05

8 Match each shape with its correct property. Shape

Exercise 4-06

Exercise 4-07

Property

a octahedron

A another name for a cube

b pyramid

B a polyhedron with triangular faces and a square base

c sphere

C has all side faces meeting at a point called the apex

d octagonal prism

D a Platonic solid with eight equilateral triangles as faces

e regular hexahedron

E has a uniform cross-section but is not a prism

f

F a polyhedron with 10 faces, 16 vertices and 24 edges

oblique prism

g square pyramid

G another name for a triangular pyramid

h cylinder

H a box shape that is slanted

i

right pyramid

I

a solid with no ﬂat faces

j

tetrahedron

J

a pyramid that is upright

9 Draw a triangular prism and count the number of: a faces b edges 10 For each of these solids, name a pair of: i intersecting edges ii parallel edges a b N T

iii skew edges c A

U

W P

R

B

V

O

NEW CENTURY MATHS 7

L

M

S J

148

c vertices

K

E

F

D

C

04 NCM7 2nd ed SB TXT.fm Page 149 Saturday, June 7, 2008 4:24 PM

11 What is a Platonic solid? Give an example of one.

Exercise 4-08

12 This solid is a regular octahedron.

Exercise 4-08

a b c d

Which two solids can be combined to make this ﬁgure? How is this octahedron regular? How many faces, vertices and edges has this octahedron? Describe Euler’s rule for convex polyhedra and show that it is true for this octahedron.

Exercise 4-06

13 Copy and complete these rectangular prisms on dot paper. a b

Exercise 4-09

14 For each of these solids, sketch the views requested.

Exercise 4-10

a

i ii iii

front view left view top view

i ii iii

top view right view front view

i ii iii

right view front view top view

ht

fro

rig

nt

b

ht

fro

rig

nt

c

front

CHAPTER 4 SOLIDS

149

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