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4

Space and geometry

Solids

The objects in our world are three-dimensional shapes or solids—cars, towers, trees, mountains, cartons, clouds, computers, flowers, oranges, even cakes and ice cream cones! Look around and you will see lots more. The Pyramid of Cheops (Khufu) at Giza in Egypt was built in the 3rd century BC. It is 153 metres high and its square base has a length of 241 metres.

In this chapter you will: ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■

understand and use the geometric terms ‘polyhedron’, ‘convex solid’, ‘uniform cross-section’, ‘base’, ‘apex’, ‘prism’, ‘pyramid’, ‘cylinder’, ‘cone’ and ‘sphere’ describe solids in terms of their faces and other properties identify right prisms and pyramids, and oblique prisms and pyramids identify right cylinders and cones, and oblique cylinders and cones construct models of polyhedra classify solids on the basis of their properties count the number of faces, vertices and edges on a polyhedron and discover Euler’s formula for convex polyhedra determine if two edges of a solid are parallel, skew or intersecting explore the history of Platonic solids and how to make them sketch, on isometric grid paper, shapes built with cubes represent solids in two dimensions from different views.

Wordbank ■ ■ ■ ■ ■ ■ ■ ■

convex solid A solid that points or curves outwards, so that a line joining any two of its faces lies completely inside it. polyhedron A solid whose faces are all flat. hexahedron Any polyhedron with six faces. cross-section A ‘slice’ of a solid, cut across it rather than along it. congruent Another word for ‘identical’. oblique Slanted, not at right angles. skew On different planes. Platonic solid Also called a ‘regular polyhedron’, a solid whose faces are all identical regular polygons.

Think! Name a solid whose faces are identical. Can you think of more than one?

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Start up Worksheet 4-01 Brainstarters 4

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.

Worksheet 4-02 A page of shapes Skillsheet 4-01 Naming shapes Geometry 4-01 The vocabulary of geometry

2 Use this link to the Geometer’s Sketchpad or Cabri Geometry activity to revise your basic geometry vocabulary. 3 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

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

c a triangle? f a pentagon?

b an octagon? e a quadrilateral?

5 What type of lines point in the same direction and never meet? 6 a Name each of the following solids:

C A

F

G

E

D

B

H

I

b Which of the solids in part a are: i prisms? ii pyramids? c Name everyday uses for each of the solids in part a. Collect some real examples and bring them to class.

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7 Which of the solids from Question 6 has: a ﬁve faces? c curved surfaces? e all faces triangular? g all faces square? i one square face? (Some have more than one answer!)

b d f h j

six vertices (corners)? no ﬂat surfaces? two triangular faces? all faces rectangular? no rectangular faces

8 Name the solid that can be created from each of these nets. a b

9 Name and describe any other solids you know. Draw an example of each one.

Naming solids In the real world, all objects are solids. Solids are three-dimensional (3-D) shapes because, not only do they have length and breadth (width), but they also have thickness (or height). 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 SOLIDS

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Convex and non-convex solids These solids are convex.

These solids are not 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.

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

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NEW CENTURY MATHS 7

d

e

f

g

h

i

2 a Copy or trace the nets below, or print out Worksheet 4-03. (For best results, paste them on cardboard.) i

ii

iii

iv

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101

Worksheet 4-03 Nets of solids 1

CHAPTER 4

v

vi

vii

b c d e

viii

Cut out each net along its boundary. Fold each net along the dotted lines to make a solid. Write the name of each solid on one of the paler faces. Unfold each net and paste it in your workbook on its dark orange face.

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

Polyhedra 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).

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face curved face

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

face

A solid whose faces are all ﬂat is called a polyhedron. The plural of ‘polyhedron’ is ‘polyhedra’ or ‘polyhedrons’. While the word ‘polygon’ in Greek means ‘many angles’ and describes any ﬂat shape with straight sides, the word ‘polyhedron’ means ‘many faces’ and describes any solid whose faces are polygons. 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), while a hexahedron has six faces (hexa = 6).

Tetrahedron (4 faces)

Hexahedron (6 faces)

This table shows the names of some polyhedra, and the number of faces they have. 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

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 solid 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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Exercise 4-02 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 c a sphere d a square pyramid e a cone 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 use a different colour to shade each pair of parallel ﬂat faces. 6 Copy and complete this table. Solid

Number of faces

Shapes of faces

Number of Number of parallel identical faces 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?

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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 103 to name these polyhedra: a b c

d

e

f

10 Why can a cube, a rectangular prism and a pentagonal pyramid all be called hexahedrons? 11 Name any solid whose faces are all identical.

Prisms and pyramids 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 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. The prism below has trapezium-shaped cross-sections, identical and parallel to its base. It is a trapezoidal prism.

base

cross-section

Trapezoidal prism SOLIDS

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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’. apex

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.

cross-section

base Triangular pyramid

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

Right pyramid

Oblique pyramid

Example 1 a b c d e f g

Draw a cross-section of this solid. Is the cross-section of this solid uniform? 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? What is the full name of this solid?

Solution a

b c d e f g

Yes, all cross-sections of this solid are identical pentagons. A prism, because the cross-section is uniform and is a polygon. Oblique because it is slanted. The base is a pentagon. The side faces are parallelograms. This solid is an oblique pentagonal prism.

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Exercise 4-03 1 Draw a cross-section of each of these solids.

Example 1

i

ii

iii

iv

v

vi

vii

viii

ix

x

xi

xii

xiii

2 Which solids in Question 1 are: a polyhedra? c pyramids?

xiv

b prisms? d neither prisms nor pyramids?

3 Describe what a prism is, in your own words. 4 Describe what a pyramid is, in your own words. 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

d

e

c

f

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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 Worksheet 4-04 Nets of solids 2

7 Name these pyramids and classify them as right pyramids or oblique pyramids. a b c

d

e

f

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

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.

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

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.

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Right cone

Oblique cone P Q

O

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 listed below. • Polyhedron: any solid whose faces are all ﬂat. • Prism: a polyhedron that has a uniform polygonal cross-section, parallel to its base. • Pyramid: a polyhedron with a polygonal base and one further vertex called the apex. • Cylinder: a solid that has a uniform circular cross-section. • Cone: a solid that has a circular base and an apex. • Sphere: a completely round solid whose surface is always the same distance from its centre. • Right prism, right pyramid, right cylinder or right cone: an upright prism, pyramid, cylinder or cone. • Oblique prism, oblique pyramid, oblique cylinder or oblique cone: a slanted prism, pyramid, cylinder or cone. The solids are shown in the table below. Category

Meaning

Polyhedron

Any solid whose faces have straight sides

Prism

A polyhedron that has a uniform cross-section along its length, identical and parallel to its base.

Right prism

An upright prism

Oblique prism

A slanted prism

Pyramid

A polyhedron with a pointed top called the apex.

Right pyramid

An upright pyramid.

Example

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Oblique pyramid

A slanted pyramid.

Platonic solid

A regular polyhedron, whose faces are identical regular polygons.

Exercise 4-04 1 For each of these solids, i write its name

ii state whether it is right, oblique or neither.

a

b

c

d

e

f

g

h

i

2 Classify each of the following solids into one or more of the given categories. (The ﬁrst one has been done for you.) • convex solid or non-convex solid • polyhedron • right prism or oblique prism • right pyramid or oblique pyramid • right cylinder or oblique cylinder • right cone or oblique cone • sphere

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A Shape A is convex and an oblique cylinder.

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

SOLIDS

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Z

AA

BB

CC

DD

EE

FF

GG

HH

II

JJ

KK

LL

MM

NN

OO

Working mathematically Applying strategies: Nets of solids

m 4c

4 cm

4 cm

4c

m 4c

m

4 cm

NEW CENTURY MATHS 7

4 cm

4 cm

4 cm

112

m

4 cm

4c

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

2 Find nets for other solids and make more models. Some can be found in Worksheet 4-04. Use this link to ﬁnd them. 3 Match the correct shape name to each net (a to l). (Some have more than one net.) triangular prism cube rectangular prism square pyramid cylinder trapezoidal prism a

b

c

d

e

f

g

h

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Worksheet 4-04 Nets of solids 2

CHAPTER 4

Worksheet 4-05 Euler’s rule

i

j

k

l

Euler’s rule Leonhard Euler (1707–1783) 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 (plural: A vertex is a corner of the solid, where edges meet. vertices) The prism in the diagram has six faces, eight vertices face and 12 edges.

Exercise 4-05 1 Count the number of faces (F), vertices (V) and edges (E) of the polyhedra you made in Exercise 4-01 and of any others you have collected. Copy and complete the table on the next page.

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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 What is Euler’s rule for convex polyhedra? Use the table from Question 1 to work it out. 3 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. 4 A piece has been sliced off each solid below. Count the number of faces (F), vertices (V) and edges (E) of each remaining solid. Does Euler’s rule still hold true for them? a

b

c

d

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

Edges of a solid The edges are the ‘lines’ of a solid, made where two faces meet. This diagram of a rectangular prism has its eight vertices labelled from A to H. We can also use these labels to describe the edges of the prism. For example, the top front edge is AD, while the back right edge is CG.

B

C

A

D

G

F E

H

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Two edges can be related in three different ways. They can be intersecting, parallel or skew. In the prism at the bottom of the previous page: • AD and CD are intersecting edges. Intersecting edges meet A at a vertex. Edges AD and CD meet at the vertex D. D

C

B

• AB and HG are parallel edges, written AB II HG. Parallel edges point in the same direction and never meet. 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 of them 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 4-06 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 C

A D F

E

2 For each of the following solids name a pair of: i intersecting edges ii parallel edges A D a b H

C

G T

L P

f

S R

V

U

O W

116

L

A

e

K

Q N

M

K

F

H

J M

O

E F

d

B

iii skew edges. N c

C

D G

B

H

J

E

A

G

NEW CENTURY MATHS 7

P

B

E

Q D

C

Q

g

R U

P

A

h

Y

D

W

T N

B I

J

V D

B

S

A

i

X

H

C

E C

H F

I

E

G

F

G

3 Copy and complete the statements below each of these solids. Choose from ‘parallel’, ‘intersecting’ or ‘skew’ to ﬁll the blanks. S

a

A

b T

R

H

G

c J

I

U W X

V

C B

L

N

Y

ST and UY are TX and RV are VW and WX are WY and TX are

K

D

M

E

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

GK and HL are MN and KL are GJ and LM are HI and IM are

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.

Perpendicular lines

Perpendicular lines

Skew 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. SOLIDS

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The Platonic solids A regular polygon is a ﬂat shape with equal sides. For example, a regular hexagon is a shape with six equal sides. A regular polyhedron or Platonic solid is a polyhedron with congruent faces. Every face is the same regular polygon and all pairs of adjacent faces make equal angles with each other. For example, a cube is a regular polyhedron because every face is a square with sides of the same length. Because a cube has six equal faces, it is also called a regular hexahedron. There are only ﬁve such Platonic solids: Platonic solid

Number of equal faces

Regular tetrahedron

tetra = 4

Regular hexahedron

hexa = 6

Regular octahedron

octa = 8

Regular dodecahedron Regular icosahedron

Tetrahedron (equilateral triangular pyramid)

Dodecahedron

dodeca = 12 icosa = 20

Hexahedron (cube)

Octahedron

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. The Greek philosopher Plato (427–327 BC) claimed that these regular polyhedra 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.

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Exercise 4-07 You can print out Worksheet 4-06 to get copies of the nets used in this Exercise.

Worksheet 4-06 Nets of Platonic solids

m 5c

5c m

Geometry 4-02 Platonic solids nets

m 5c

5c

m

5c

5 cm

m

m

5 cm 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?

5 cm

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.

m

5c

m

5 cm

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?

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4c

m 4c

m

3 A regular icosahedron is shown on the right. a Draw and cut out the net of a regular icosahedron below, with side lengths of 5 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?

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

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5 A regular dodecahedron is shown on the right. a Draw and cut out this net of a regular dodecahedron with side lengths of 5 cm, then make it.

3 cm

3 cm

3 cm

3c

m

b c d e f g

m

3c

How many faces has the regular dodecahedron? What is the shape of each face? How many edges does it have? How many vertices does it have? Is Euler’s rule true for this dodecahedron? Does this dodecahedron have any: i parallel edges? ii skew edges?

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

Description

Shape of each face

Number of faces

Two square pyramids joined at bases Flattened soccer ball pattern

Regular pentagon

Crystal-shaped

20 4 Square

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

b is a pyramid? e is a prism?

c has 12 edges? f has pentagonal faces? SOLIDS

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Working mathematically 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 or in a project book. 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.

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. Horizontal edges are drawn horizontally on the paper.

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. In both cases, the parallel edges of the solid must be drawn as parallel on the paper.

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Exercise 4-08 For this exercise you will need square dot paper, isometric dot paper and construction cubes. Both types of dot paper can be printed out. Use these links to ﬁnd them. 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 2 Use cubes to make each of the following solids, then draw them on isometric dot paper. a

b

e

f

c

d

Geometry 4-03 Square grid Geometry 4-04 Isometric grid Worksheet Appendix 8 Square dot paper Worksheet Appendix 9 Isometric dot paper

g

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.

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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

d

6 cubes

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

124

ii

NEW CENTURY MATHS 7

d D? iii

:

iv

v

vi

vii

viii

x ix

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

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

B

D

C

E

Working mathematically Applying strategies: Drawing prisms and pyramids 1 Try drawing solids without using dot paper. Use a ruler and pencil to practise drawing the following solids: • a cube • a square pyramid • a rectangular prism • a cylinder 2 There are tricks you can use to draw these so that they ‘look right’. Find out what these tricks are and use them to improve your drawings.

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Different views of solids Example 2 For this solid, draw: a the front view b the left view c the top view. left

nt

fro

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

Example 3 For this solid, draw: a the front view b the top view.

Solution a

126

front

b

NEW CENTURY MATHS 7

Exercise 4-09 1 For each of these solids, draw the views requested. a b

fro

nt

Example 2

left

ht

rig

i top view ii right view iii front view

nt

fro

i left view ii back view iii top view

c

d

fro

nt

left

ht

rig

i back view ii top view iii right view

nt

fro

i right view ii front view iii top view

e

f

fro

nt

left

ht

rig

i front view ii left view iii top view

nt

fro

i back view ii right view iii top view

g

h

nt

left

i top view ii front view iii left view

fro

ht

rig

fro

nt

i right view ii back view iii top view SOLIDS

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

front

front

i front view ii top view

i front view ii right view iii top view d

nt

c

fro

Example 3

front

i left view ii top view iii front view e

i top view ii front view f

front

i ii iii iv

front view left view right view top view

front

i top view ii front view

3 Draw each of these solids from the top, front and side views. a b

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c

d

4 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

top

side

front

side

c 5 cubes

top

front

side

top

front

side

top

front

side

top

front

side

d 5 cubes

e 7 cubes

f

8 cubes

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Power plus 1 Draw: 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. The following 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. vertex region

edg

e

This network has 3 regions, 5 vertices This network has 3 regions, 4 vertices and 6 edges. and 5 edges. 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

g

h

i

j

k

l

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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 the following 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

Worksheet 4-07 Solids crossword

Language of maths apex congruent cube edge hexahedron oblique prism parallel edges polygon pyramid right prism slant tetrahedron

base convex cylinder Euler’s rule intersecting edges oblique pyramid plane polyhedron/polyhedra regular polygon right pyramid solid uniform

cone cross-section dodecahedron face icosahedron octahedron Platonic solid prism regular polyhedron skew edges sphere vertex/vertices

1 Find the ﬁve Platonic solids in the list above. 2 How many faces has a tetrahedron? 3 What is the difference between the base of a prism and the base of a pyramid? 4 What is a ‘polyhedron’? Give an example of a polyhedron and an example that is not a polyhedron. 5 What type of polyhedra has a uniform cross-section? 6 What is the difference between a ‘right’ prism and an ‘oblique’ prism? 7 What do the Greek preﬁxes ‘hexa’ and ‘poly’ mean? 8 What is the difference between a convex polygon and a non-convex polygon?

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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

Solids with curved surfaces

Drawing and building solids

S O L IDS Polyhedra Platonic solids

Pyramids

Prisms

Right Euler’s rule

Right

Oblique

Faces

Vertices

Edges

Parallel

Intersecting

Skew

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Oblique

Chapter 4

Review

Topic test Chapter 4

1 Copy and complete the table below for the three solids shown. A

B

Ex 4-02

C

Solid A

Solid B

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?

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

Ex 4-03

c a rectangular prism

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

d

e

f

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

d

e

Ex 4-03

Ex 4-03

f

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Ex 4-01

5 Which solids from Questions 1, 3 and 4 are non-convex?

Ex 4-04

6 Match each shape with its correct property. Shape

Ex 4-05

Ex 4-06

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

7 Draw a triangular prism and count the number of: a faces b edges

c vertices

8 For each of these solids, name a pair of: i intersecting edges ii parallel edges a b N T

c

iii skew edges

U

W P

R

V

O

Ex 4-07

Ex 4-08

L

M

S J

Ex 4-07

K

9 What is a Platonic solid? Give an example of one. 10 This solid is a regular octahedron. a Which two solids can be combined to make this ﬁgure? b How is this octahedron regular? c How many faces, vertices and edges has this octahedron? d Describe Euler’s rule for convex polyhedra and show that it is true for this octahedron. 11 Copy and complete these rectangular prisms on dot paper. a b

134

B

A

NEW CENTURY MATHS 7

E

F

D

C

12 For each of these solids, sketch the views requested. a i front view ii left view iii top view

Ex 4-09

fro

nt

b

i top view ii right view iii front view

ht

fro

rig

nt

c

i right view ii front view iii top view front

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View more...
Space and geometry

Solids

The objects in our world are three-dimensional shapes or solids—cars, towers, trees, mountains, cartons, clouds, computers, flowers, oranges, even cakes and ice cream cones! Look around and you will see lots more. The Pyramid of Cheops (Khufu) at Giza in Egypt was built in the 3rd century BC. It is 153 metres high and its square base has a length of 241 metres.

In this chapter you will: ■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■

understand and use the geometric terms ‘polyhedron’, ‘convex solid’, ‘uniform cross-section’, ‘base’, ‘apex’, ‘prism’, ‘pyramid’, ‘cylinder’, ‘cone’ and ‘sphere’ describe solids in terms of their faces and other properties identify right prisms and pyramids, and oblique prisms and pyramids identify right cylinders and cones, and oblique cylinders and cones construct models of polyhedra classify solids on the basis of their properties count the number of faces, vertices and edges on a polyhedron and discover Euler’s formula for convex polyhedra determine if two edges of a solid are parallel, skew or intersecting explore the history of Platonic solids and how to make them sketch, on isometric grid paper, shapes built with cubes represent solids in two dimensions from different views.

Wordbank ■ ■ ■ ■ ■ ■ ■ ■

convex solid A solid that points or curves outwards, so that a line joining any two of its faces lies completely inside it. polyhedron A solid whose faces are all flat. hexahedron Any polyhedron with six faces. cross-section A ‘slice’ of a solid, cut across it rather than along it. congruent Another word for ‘identical’. oblique Slanted, not at right angles. skew On different planes. Platonic solid Also called a ‘regular polyhedron’, a solid whose faces are all identical regular polygons.

Think! Name a solid whose faces are identical. Can you think of more than one?

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Start up Worksheet 4-01 Brainstarters 4

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.

Worksheet 4-02 A page of shapes Skillsheet 4-01 Naming shapes Geometry 4-01 The vocabulary of geometry

2 Use this link to the Geometer’s Sketchpad or Cabri Geometry activity to revise your basic geometry vocabulary. 3 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

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

c a triangle? f a pentagon?

b an octagon? e a quadrilateral?

5 What type of lines point in the same direction and never meet? 6 a Name each of the following solids:

C A

F

G

E

D

B

H

I

b Which of the solids in part a are: i prisms? ii pyramids? c Name everyday uses for each of the solids in part a. Collect some real examples and bring them to class.

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7 Which of the solids from Question 6 has: a ﬁve faces? c curved surfaces? e all faces triangular? g all faces square? i one square face? (Some have more than one answer!)

b d f h j

six vertices (corners)? no ﬂat surfaces? two triangular faces? all faces rectangular? no rectangular faces

8 Name the solid that can be created from each of these nets. a b

9 Name and describe any other solids you know. Draw an example of each one.

Naming solids In the real world, all objects are solids. Solids are three-dimensional (3-D) shapes because, not only do they have length and breadth (width), but they also have thickness (or height). 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 SOLIDS

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CHAPTER 4

Convex and non-convex solids These solids are convex.

These solids are not 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.

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

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NEW CENTURY MATHS 7

d

e

f

g

h

i

2 a Copy or trace the nets below, or print out Worksheet 4-03. (For best results, paste them on cardboard.) i

ii

iii

iv

SOLIDS

101

Worksheet 4-03 Nets of solids 1

CHAPTER 4

v

vi

vii

b c d e

viii

Cut out each net along its boundary. Fold each net along the dotted lines to make a solid. Write the name of each solid on one of the paler faces. Unfold each net and paste it in your workbook on its dark orange face.

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

Polyhedra 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).

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face curved face

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

face

A solid whose faces are all ﬂat is called a polyhedron. The plural of ‘polyhedron’ is ‘polyhedra’ or ‘polyhedrons’. While the word ‘polygon’ in Greek means ‘many angles’ and describes any ﬂat shape with straight sides, the word ‘polyhedron’ means ‘many faces’ and describes any solid whose faces are polygons. 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), while a hexahedron has six faces (hexa = 6).

Tetrahedron (4 faces)

Hexahedron (6 faces)

This table shows the names of some polyhedra, and the number of faces they have. 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

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 solid 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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Exercise 4-02 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 c a sphere d a square pyramid e a cone 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 use a different colour to shade each pair of parallel ﬂat faces. 6 Copy and complete this table. Solid

Number of faces

Shapes of faces

Number of Number of parallel identical faces 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?

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NEW CENTURY MATHS 7

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 103 to name these polyhedra: a b c

d

e

f

10 Why can a cube, a rectangular prism and a pentagonal pyramid all be called hexahedrons? 11 Name any solid whose faces are all identical.

Prisms and pyramids 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 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. The prism below has trapezium-shaped cross-sections, identical and parallel to its base. It is a trapezoidal prism.

base

cross-section

Trapezoidal prism SOLIDS

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CHAPTER 4

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’. apex

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.

cross-section

base Triangular pyramid

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

Right pyramid

Oblique pyramid

Example 1 a b c d e f g

Draw a cross-section of this solid. Is the cross-section of this solid uniform? 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? What is the full name of this solid?

Solution a

b c d e f g

Yes, all cross-sections of this solid are identical pentagons. A prism, because the cross-section is uniform and is a polygon. Oblique because it is slanted. The base is a pentagon. The side faces are parallelograms. This solid is an oblique pentagonal prism.

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NEW CENTURY MATHS 7

Exercise 4-03 1 Draw a cross-section of each of these solids.

Example 1

i

ii

iii

iv

v

vi

vii

viii

ix

x

xi

xii

xiii

2 Which solids in Question 1 are: a polyhedra? c pyramids?

xiv

b prisms? d neither prisms nor pyramids?

3 Describe what a prism is, in your own words. 4 Describe what a pyramid is, in your own words. 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

d

e

c

f

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 Worksheet 4-04 Nets of solids 2

7 Name these pyramids and classify them as right pyramids or oblique pyramids. a b c

d

e

f

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

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.

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

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.

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NEW CENTURY MATHS 7

Right cone

Oblique cone P Q

O

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 listed below. • Polyhedron: any solid whose faces are all ﬂat. • Prism: a polyhedron that has a uniform polygonal cross-section, parallel to its base. • Pyramid: a polyhedron with a polygonal base and one further vertex called the apex. • Cylinder: a solid that has a uniform circular cross-section. • Cone: a solid that has a circular base and an apex. • Sphere: a completely round solid whose surface is always the same distance from its centre. • Right prism, right pyramid, right cylinder or right cone: an upright prism, pyramid, cylinder or cone. • Oblique prism, oblique pyramid, oblique cylinder or oblique cone: a slanted prism, pyramid, cylinder or cone. The solids are shown in the table below. Category

Meaning

Polyhedron

Any solid whose faces have straight sides

Prism

A polyhedron that has a uniform cross-section along its length, identical and parallel to its base.

Right prism

An upright prism

Oblique prism

A slanted prism

Pyramid

A polyhedron with a pointed top called the apex.

Right pyramid

An upright pyramid.

Example

SOLIDS

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CHAPTER 4

Oblique pyramid

A slanted pyramid.

Platonic solid

A regular polyhedron, whose faces are identical regular polygons.

Exercise 4-04 1 For each of these solids, i write its name

ii state whether it is right, oblique or neither.

a

b

c

d

e

f

g

h

i

2 Classify each of the following solids into one or more of the given categories. (The ﬁrst one has been done for you.) • convex solid or non-convex solid • polyhedron • right prism or oblique prism • right pyramid or oblique pyramid • right cylinder or oblique cylinder • right cone or oblique cone • sphere

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NEW CENTURY MATHS 7

A Shape A is convex and an oblique cylinder.

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

SOLIDS

111

CHAPTER 4

Z

AA

BB

CC

DD

EE

FF

GG

HH

II

JJ

KK

LL

MM

NN

OO

Working mathematically Applying strategies: Nets of solids

m 4c

4 cm

4 cm

4c

m 4c

m

4 cm

NEW CENTURY MATHS 7

4 cm

4 cm

4 cm

112

m

4 cm

4c

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

2 Find nets for other solids and make more models. Some can be found in Worksheet 4-04. Use this link to ﬁnd them. 3 Match the correct shape name to each net (a to l). (Some have more than one net.) triangular prism cube rectangular prism square pyramid cylinder trapezoidal prism a

b

c

d

e

f

g

h

SOLIDS

113

Worksheet 4-04 Nets of solids 2

CHAPTER 4

Worksheet 4-05 Euler’s rule

i

j

k

l

Euler’s rule Leonhard Euler (1707–1783) 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 (plural: A vertex is a corner of the solid, where edges meet. vertices) The prism in the diagram has six faces, eight vertices face and 12 edges.

Exercise 4-05 1 Count the number of faces (F), vertices (V) and edges (E) of the polyhedra you made in Exercise 4-01 and of any others you have collected. Copy and complete the table on the next page.

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NEW CENTURY MATHS 7

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 What is Euler’s rule for convex polyhedra? Use the table from Question 1 to work it out. 3 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. 4 A piece has been sliced off each solid below. Count the number of faces (F), vertices (V) and edges (E) of each remaining solid. Does Euler’s rule still hold true for them? a

b

c

d

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

Edges of a solid The edges are the ‘lines’ of a solid, made where two faces meet. This diagram of a rectangular prism has its eight vertices labelled from A to H. We can also use these labels to describe the edges of the prism. For example, the top front edge is AD, while the back right edge is CG.

B

C

A

D

G

F E

H

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Two edges can be related in three different ways. They can be intersecting, parallel or skew. In the prism at the bottom of the previous page: • AD and CD are intersecting edges. Intersecting edges meet A at a vertex. Edges AD and CD meet at the vertex D. D

C

B

• AB and HG are parallel edges, written AB II HG. Parallel edges point in the same direction and never meet. 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 of them 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 4-06 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 C

A D F

E

2 For each of the following solids name a pair of: i intersecting edges ii parallel edges A D a b H

C

G T

L P

f

S R

V

U

O W

116

L

A

e

K

Q N

M

K

F

H

J M

O

E F

d

B

iii skew edges. N c

C

D G

B

H

J

E

A

G

NEW CENTURY MATHS 7

P

B

E

Q D

C

Q

g

R U

P

A

h

Y

D

W

T N

B I

J

V D

B

S

A

i

X

H

C

E C

H F

I

E

G

F

G

3 Copy and complete the statements below each of these solids. Choose from ‘parallel’, ‘intersecting’ or ‘skew’ to ﬁll the blanks. S

a

A

b T

R

H

G

c J

I

U W X

V

C B

L

N

Y

ST and UY are TX and RV are VW and WX are WY and TX are

K

D

M

E

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

GK and HL are MN and KL are GJ and LM are HI and IM are

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.

Perpendicular lines

Perpendicular lines

Skew 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. SOLIDS

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CHAPTER 4

The Platonic solids A regular polygon is a ﬂat shape with equal sides. For example, a regular hexagon is a shape with six equal sides. A regular polyhedron or Platonic solid is a polyhedron with congruent faces. Every face is the same regular polygon and all pairs of adjacent faces make equal angles with each other. For example, a cube is a regular polyhedron because every face is a square with sides of the same length. Because a cube has six equal faces, it is also called a regular hexahedron. There are only ﬁve such Platonic solids: Platonic solid

Number of equal faces

Regular tetrahedron

tetra = 4

Regular hexahedron

hexa = 6

Regular octahedron

octa = 8

Regular dodecahedron Regular icosahedron

Tetrahedron (equilateral triangular pyramid)

Dodecahedron

dodeca = 12 icosa = 20

Hexahedron (cube)

Octahedron

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. The Greek philosopher Plato (427–327 BC) claimed that these regular polyhedra 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.

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NEW CENTURY MATHS 7

Exercise 4-07 You can print out Worksheet 4-06 to get copies of the nets used in this Exercise.

Worksheet 4-06 Nets of Platonic solids

m 5c

5c m

Geometry 4-02 Platonic solids nets

m 5c

5c

m

5c

5 cm

m

m

5 cm 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?

5 cm

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.

m

5c

m

5 cm

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?

SOLIDS

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CHAPTER 4

4c

m 4c

m

3 A regular icosahedron is shown on the right. a Draw and cut out the net of a regular icosahedron below, with side lengths of 5 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?

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

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5 A regular dodecahedron is shown on the right. a Draw and cut out this net of a regular dodecahedron with side lengths of 5 cm, then make it.

3 cm

3 cm

3 cm

3c

m

b c d e f g

m

3c

How many faces has the regular dodecahedron? What is the shape of each face? How many edges does it have? How many vertices does it have? Is Euler’s rule true for this dodecahedron? Does this dodecahedron have any: i parallel edges? ii skew edges?

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

Description

Shape of each face

Number of faces

Two square pyramids joined at bases Flattened soccer ball pattern

Regular pentagon

Crystal-shaped

20 4 Square

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

b is a pyramid? e is a prism?

c has 12 edges? f has pentagonal faces? SOLIDS

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Working mathematically 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 or in a project book. 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.

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. Horizontal edges are drawn horizontally on the paper.

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. In both cases, the parallel edges of the solid must be drawn as parallel on the paper.

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Exercise 4-08 For this exercise you will need square dot paper, isometric dot paper and construction cubes. Both types of dot paper can be printed out. Use these links to ﬁnd them. 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 2 Use cubes to make each of the following solids, then draw them on isometric dot paper. a

b

e

f

c

d

Geometry 4-03 Square grid Geometry 4-04 Isometric grid Worksheet Appendix 8 Square dot paper Worksheet Appendix 9 Isometric dot paper

g

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.

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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

d

6 cubes

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

124

ii

NEW CENTURY MATHS 7

d D? iii

:

iv

v

vi

vii

viii

x ix

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

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

B

D

C

E

Working mathematically Applying strategies: Drawing prisms and pyramids 1 Try drawing solids without using dot paper. Use a ruler and pencil to practise drawing the following solids: • a cube • a square pyramid • a rectangular prism • a cylinder 2 There are tricks you can use to draw these so that they ‘look right’. Find out what these tricks are and use them to improve your drawings.

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Different views of solids Example 2 For this solid, draw: a the front view b the left view c the top view. left

nt

fro

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

Example 3 For this solid, draw: a the front view b the top view.

Solution a

126

front

b

NEW CENTURY MATHS 7

Exercise 4-09 1 For each of these solids, draw the views requested. a b

fro

nt

Example 2

left

ht

rig

i top view ii right view iii front view

nt

fro

i left view ii back view iii top view

c

d

fro

nt

left

ht

rig

i back view ii top view iii right view

nt

fro

i right view ii front view iii top view

e

f

fro

nt

left

ht

rig

i front view ii left view iii top view

nt

fro

i back view ii right view iii top view

g

h

nt

left

i top view ii front view iii left view

fro

ht

rig

fro

nt

i right view ii back view iii top view SOLIDS

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CHAPTER 4

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

front

front

i front view ii top view

i front view ii right view iii top view d

nt

c

fro

Example 3

front

i left view ii top view iii front view e

i top view ii front view f

front

i ii iii iv

front view left view right view top view

front

i top view ii front view

3 Draw each of these solids from the top, front and side views. a b

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NEW CENTURY MATHS 7

c

d

4 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

top

side

front

side

c 5 cubes

top

front

side

top

front

side

top

front

side

top

front

side

d 5 cubes

e 7 cubes

f

8 cubes

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Power plus 1 Draw: 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. The following 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. vertex region

edg

e

This network has 3 regions, 5 vertices This network has 3 regions, 4 vertices and 6 edges. and 5 edges. 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

g

h

i

j

k

l

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NEW CENTURY MATHS 7

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 the following 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

Worksheet 4-07 Solids crossword

Language of maths apex congruent cube edge hexahedron oblique prism parallel edges polygon pyramid right prism slant tetrahedron

base convex cylinder Euler’s rule intersecting edges oblique pyramid plane polyhedron/polyhedra regular polygon right pyramid solid uniform

cone cross-section dodecahedron face icosahedron octahedron Platonic solid prism regular polyhedron skew edges sphere vertex/vertices

1 Find the ﬁve Platonic solids in the list above. 2 How many faces has a tetrahedron? 3 What is the difference between the base of a prism and the base of a pyramid? 4 What is a ‘polyhedron’? Give an example of a polyhedron and an example that is not a polyhedron. 5 What type of polyhedra has a uniform cross-section? 6 What is the difference between a ‘right’ prism and an ‘oblique’ prism? 7 What do the Greek preﬁxes ‘hexa’ and ‘poly’ mean? 8 What is the difference between a convex polygon and a non-convex polygon?

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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

Solids with curved surfaces

Drawing and building solids

S O L IDS Polyhedra Platonic solids

Pyramids

Prisms

Right Euler’s rule

Right

Oblique

Faces

Vertices

Edges

Parallel

Intersecting

Skew

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NEW CENTURY MATHS 7

Oblique

Chapter 4

Review

Topic test Chapter 4

1 Copy and complete the table below for the three solids shown. A

B

Ex 4-02

C

Solid A

Solid B

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?

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

Ex 4-03

c a rectangular prism

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

d

e

f

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

d

e

Ex 4-03

Ex 4-03

f

SOLIDS

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CHAPTER 4

Ex 4-01

5 Which solids from Questions 1, 3 and 4 are non-convex?

Ex 4-04

6 Match each shape with its correct property. Shape

Ex 4-05

Ex 4-06

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

7 Draw a triangular prism and count the number of: a faces b edges

c vertices

8 For each of these solids, name a pair of: i intersecting edges ii parallel edges a b N T

c

iii skew edges

U

W P

R

V

O

Ex 4-07

Ex 4-08

L

M

S J

Ex 4-07

K

9 What is a Platonic solid? Give an example of one. 10 This solid is a regular octahedron. a Which two solids can be combined to make this ﬁgure? b How is this octahedron regular? c How many faces, vertices and edges has this octahedron? d Describe Euler’s rule for convex polyhedra and show that it is true for this octahedron. 11 Copy and complete these rectangular prisms on dot paper. a b

134

B

A

NEW CENTURY MATHS 7

E

F

D

C

12 For each of these solids, sketch the views requested. a i front view ii left view iii top view

Ex 4-09

fro

nt

b

i top view ii right view iii front view

ht

fro

rig

nt

c

i right view ii front view iii top view front

SOLIDS

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CHAPTER 4

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