Polygons and polyhedra - WordPress.com · Polygons and polyhedra. 422 Maths Quest 7 for Victoria Introduction The world around you is ﬁlled with many different shapes and objects

11

Val and Peter want to replace their front gate with another of the same design. What shapes are formed by the metal bars of the gate? To have this gate made, they need to supply a diagram of it with all measurements and angles shown. In this chapter, you will look at different shapes and their properties, including angles.

Polygons andpolyhedra

422

M a t h s Q u e s t 7 f o r V i c t o r i a

Introduction

The world around you is filled with many different shapes and objects. The roof of thishouse in Falls Creek is shaped as a triangle when viewed from the front. This lets thewinter snow slide off the roof.

Dice are shaped as cubes so that each of the 6 numbers are equally likely to appear on the uppermost face.

This soccer ball is made up of five-sided andsix-sided shapes that almost form a spherewhich can be rolled and kicked along theground.

As you learn about the properties ofshapes and objects you will understandhow and why they are used in the worldaround you.

In this chapter you will learn the names and properties of many common

2-dimensional shapes, called

polygons

, and3-dimensional objects, called

polyhedra

. Youwill also learn to construct 3-dimensionalobjects and draw them on a page.

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a

423

Types of triangles

The word triangle means 3 angles. Every triangle has 3 angles and 3 sides. Capital letters of the English alphabet at the

vertices of triangles can be used to identify triangles. When identifying triangles, the vertices are listed in either a clockwise or anticlockwise direction, beginning with any vertex. Instead of

the word ‘triangle’ the symbol

L

is used. For example, the triangle shown at right can be referred to as

L

ABC. However, it would be equally appropriate to name it

L

BCA,

L

CAB,

L

ACB,

L

BAC or

L

CBA. Triangles can be classified according to the length of their sides or the size of their

angles.

Classifying triangles according to the length of their sides

An

equilateral triangle

has all sides equal in length.

Note that identical marks on the sides of a triangle are used to indicate that the sides have the same length. The angles of an equilateral triangle are equal in size. This is shown by placing identical curves on each angle.

An

isosceles triangle

has 2 sides of equal length.

The side of the isosceles triangle that has a different length, is often called the base of the triangle. The angles adjacent to the base of the isosceles triangle are equal in size. On the diagram at right, the side markings show the 2 sides that are equal and the angle markings show the 2 angles that are equal.

A

scalene triangle

has no equal sides.

The different side markings on the diagram show that the 3 sides have different lengths. A scalene triangle has all 3 angles of different size. This is shown by different angle markings.

B

A C

424

M a t h s Q u e s t 7 f o r V i c t o r i a

Classifying triangles according to the size of their angles

A

right-angled triangle

has one of its angles equal to 90

°

(that is, one of its angles is a right angle).

On the diagram, putting a small square in the corner marks the right angle.

An

acute-angled triangle

has all angles smaller than 90

°

(that is, all 3 angles are acute).

An

obtuse-angled triangle

has 1 angle greater than 90

°

(that is, one angle is obtuse).

Classify each of these triangles according to the lengths of their sides.a b c

THINK WRITE

a Sides AB and AC have identical markings on them, which indicates that they are of equal length. So LABC has 2 equal sides. Classify it accordingly.

a LABC is an isosceles triangle.

b The 3 sides of LMNP have identical markings on them, which means that all 3 sides are equal in length. Classify this triangle.

b LMNP is an equilateral triangle.

c All 3 sides of LPRS are marked differently. Therefore, no sides in this triangle are equal in length. Use this information to classify the triangle.

c LPRS is a scalene triangle.

B

CA

N

PM P

R

S

1WORKEDExample


425

Types of triangles

1

Classify each of these triangles according to the lengths of their sides.

a b c

d e f

Classify each of the triangles in worked example 1 according to the size of their angles.

THINK WRITE

a In LABC, ∠CAB is marked as the right angle, so classify it accordingly.

a LABC is a right-angled triangle.

b In LMNP all angles are less than 90°, so classify this triangle.

b LMNP is an acute-angled triangle.

c In LPRS, ∠PRS is greater than 90°; that is, it is obtuse. Use this information to classify the triangle.

c LPRS is an obtuse-angled triangle.

2WORKEDExample

remember1. According to the lengths of the sides, a triangle can be classified as being:

(a) equilateral (3 equal sides)(b) isosceles (2 equal sides)(c) scalene (no equal sides).

2. A triangle can be classified according to the angle size, as being: (a) acute-angled (all 3 angles are acute)(b) right-angled (1 angle is a right angle)(c) obtuse-angled (1 angle is obtuse).

remember

11A11.1

Classifyingtriangles(sides)

WORKEDExample

1

426 M a t h s Q u e s t 7 f o r V i c t o r i a

2 Classify each of the triangles in question 1 according to the size of their angles.

3 Add side and angle markings to these diagrams to show that:a b

LRST is an equilateral triangle LUVW is an isosceles triangle

c d

LPQR is a scalene triangle LMNP is a right-angled triangle

e f

LABC is a right-angled and LMNO is a right-angled and isosceles triangle scalene triangle.

4

a Which of these triangles is an equilateral triangle? A B C

D E

Classifying triangles (angles)

WORKEDExample

2

S

TR

V

WU

Q

R

P N

PM

B

CA

M N

O

mmultiple choiceultiple choice

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 427b Which of these triangles is not a scalene triangle?

A B C

D E

5

a Which of these triangles is both right-angled and scalene?

A B C

D E

b Which of these triangles is both acute-angled and isosceles?

A B C

D E



6 What types of triangles can you see in this picture?

7 Write down 3 acute triangles you can see around you.

8 Find one example in your classroom or home of each of the 6 types of triangles described in this chapter. Describe clearly where the triangle occurs, draw the triangle and classify it according to both side and angle types.

9 In the picture at right:

a how many equilateral triangles can you find?

b how many right-angled triangles can you find?

c how many isosceles triangles can you find?

10 Use your ruler, pencil and protractor to accurately draw:a an equilateral triangle with side lengths 6 cm and all angles 60° b an isosceles triangle with two sides which are 6 cm each with a 40° angle between

themc a right-angled triangle whose two short sides are 6 cm and 8 cm.

How long is the longer side?d a scalene triangle with two of the sides measuring 4 cm and 5 cm and an angle of

70° between the two sides.

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 How many triangles can you find in these shapes? a b

2 How many triangles can you find in these shapes? a b

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 429

Angles in a triangle

As a result of this investigation you should have discovered the rule that is stated below.

It can be shown that the sum of the 3 angles in any triangle is equal to 180°.

In the triangle at right, a + b + c = 180° where the3 angles of the triangle are a, b and c

This rule can be used to find missing angles in triangles, as shown in the followingexamples.

Sum of angles in a triangleYou will need: a ruler and a protractor.

1. Draw an acute-angled triangle in your workbook.

2. Use a protractor to measure each of the 3 angles.

3. Find the sum of the 3 angles.

4. Draw up a table like the one shown below and write in your results.

5. Repeat steps 1–3 for the 4 other triangles in the table.

6. Write down any patterns that you have observed in relation to the sum of the angles in a triangle.

Triangle First angleSecond angle Third angle

Sum of angles

1. Acute-angled

2. Obtuse-angled

3. Right-angled

4. Isosceles

5. Scalene

Angle sumof a

triangle

b

ac


In the previous section it was discussed that the angles at the base of anisosceles triangle are equal in size. Worked examples 4 and 5 illustrate the use ofthis property.

Find the value of the pronumeral in this triangle.

THINK WRITE

The sum of the 3 angles (b, 35° and 58°) must be 180°. Write this as an equation.

b + 35° + 58° = 180°

Simplify by adding 35° and 58° together.

b + 93° = 180°

Use inspection or backtracking to solve for b.

b = 180° − 93°b = 87°

b

35º

58º

1

2

3

+ 93°

– 93°

b + 93°

180°

b

87°

3WORKEDExample

Find the value of the pronumeral in the following triangle.

THINK WRITE

The markings on the diagram indicate that LABC is isosceles with AB = BC. Therefore, the angles at the base are equal in size; that is, ∠BCA = ∠BAC = 74°.

∠BAC = 74°

All 3 angles in a triangle must add up to 180°.

∠ABC + ∠BAC + ∠BCA = 180°h + 74° + 74° = 180°

Simplify. h + 148° = 180°Solve for h. h = 180° − 148°

h = 32°

74º

B

A

C

h

1

2

34

4WORKEDExample


Interior and exterior angles of a triangleThe angles inside a triangle are called interior angles. If any side of a triangle isextended outwards, the angle formed is called an exterior angle. The exterior angle andthe interior angle adjacent to it are supplementary and therefore add up to 180°.

Find the value of the pronumeral in the following triangle.

THINK WRITE

From the diagram we can see that LMNP is isosceles with MN = NP. Hence, ∠NPM = ∠NMP = a.

∠NPM = a

Form an equation by putting the sum of the angles on one side and 180° on the other side of the equals sign.

∠NMP + ∠NPM + ∠MNP = 180°a + a + 40° = 180°

Simplify by collecting like terms. 2a + 40° = 180°

Use inspection or backtracking to solve for a.

2a = 180° − 40°2a = 140°

a =

a = 70°

40º

N

PMa

1

2

3

4

× 2

÷ 2

2aa

70°

+ 40

– 40

2a + 40°

180°140°

140°2

-----------

5WORKEDExample

MQ Vic 7 fig 11-55

B

CA D

interiorangles

exteriorangles

HACB + HBCD = 180º

Interior angle

Exterior angle


Angles in a triangle

1 Find the value of the pronumeral in each of the following triangles.a b c

d e f

Find the value of the pronumerals in the diagram at right.

THINK WRITE

∠BAC (angle p) together with its adjacent exterior angle (∠DAB) add up to 180°. Furthermore, ∠DAB = 125°. So form an equation and solve for p.

∠BAC = p; ∠DAB = 125°;∠BAC + ∠DAB = 180°So p + 125° = 180°.

p = 180° − 125°p = 55°

The interior angles of LABC add up to 180°. Identify the values of the angles and form an equation.

∠BCA + ∠BAC + ∠ABC = 180°∠BCA = 83° ∠BAC = p = 55°

∠ABC = nSo 83° + 55° + n = 180°.

Simplify by adding 83° and 55° and then solve for n.

n + 138° = 180°n = 180° − 138°n = 42°

n

p

A

B

CD83º125º

1

2

3

6WORKEDExample

remember1. The sum of the interior angles in any triangle is equal to 180°.2. The angles at the base of an isosceles triangle are equal in size.3. An exterior angle of a triangle, and an interior angle adjacent to it, are

supplementary (that is, add up to 180°).

remember

11B11.2

Angle sum of a triangle

Triangles

WORKEDExample

355º

68º x

25º

30º

g

96º

40º

t

60º

60ºk

54º

f

33º 30º

60ºz

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 4332 Find the value of the pronumeral in each of the following right-angled triangles.

a b c

3 Find the value of the pronumeral in each of the following triangles.

a b c

4 Find the value of the pronumeral in each of the following triangles.

a b c

d e f

5 Find the missing angle in each of the following diagrams.

a b

45º

d

b

25º

40ºa

WORKEDExample

4 c

55º e

52º64º

n

WORKEDExample

5

48º

u k

28º

d

32º

t57º

f

70º p

b70°

p

60°


c d

6 a An isosceles triangle has 2 angles of 55° each. Find the size of the third angle.b An isosceles triangle has 2 angles of 12° each. Find the size of the third angle.c Two angles of a triangle are 55° and 75° respectively. Find the third angle.d Two angles of a triangle are 48° and 68° respectively. Find the third angle.

7 Find the value of the pronumerals in each of the following diagrams.a b

c d

e f

g h

8 a Use a ruler and a protractor to construct each of the following triangles.i An isosceles triangle with a base of 4 cm and equal angles of 50° each.ii An isosceles triangle with two sides which are 5 cm each and two equal angles

which are 45° each. b On your diagrams label the size of each angle. Classify the triangles according to

the size of their angles.

9 Below are sets of 3 angles. For each set state whether or not it is possible to constructa triangle with these angles. Give a reason for your answer.a 40°, 40°, 100° b 45°, 60°, 70°c 45°, 55°, 85° d 111°, 34.5°, 34.5°

10 Explain in your own words why it is impossible to construct a triangle with 2 obtuse angles.

100°50°

k

p

62° 62°

WWORKEDORKEDEExamplexample

6

60º130º

n

p130º158º a

b

130º

50º

x y

125º

t

s

72º26º

b55º

34º

n

120º

m

m 56º

t

t

11.1


1 Name the triangle shown at right based on the length of the sides.

2 Name the triangle in question 1 based on the size of its angles.

3 A triangle has angles of 35° and 76°. Find the third angle.

4 One angle of a right-angled triangle is 37°. Find the third angle.

In questions 5 to 9, find the missing angle in each triangle.

5 6 7

8 9

10 Find the missing angle shown in the photograph.

1

72º

57º

j

38º

a

48º

h

64º

y

22º

118º

t

x

41°


Types of quadrilateralsAny 2-dimensional closed shape with 4 straight sides is called a quadrilateral. Quad-means four, as in quadruplets (four babies), or quadriplegic (paralysed in all fourlimbs). Lateral means sides, as in lateral movement (sideways movement) or lateralthinking (thinking sideways, or around, a problem).

All quadrilaterals can be divided into 2 major groups: parallelograms and otherquadrilaterals.

Parallelograms are quadrilaterals with both pairs of opposite sides being parallel to each other. Parallelograms include rectangles, squares and rhombuses (diamonds).

The table below shows different parallelograms and their properties. Note thatparallel sides are marked with identical arrows.

Other quadrilaterals include trapeziums, kites and irregular quadrilaterals. Thefollowing table shows properties of these shapes.

Parallelogram Shape Properties

Parallelogram Opposite sides are equal in length.

Opposite angles are equal in size.

Rectangle Opposite sides are equal in length.

All angles are the same and equal to 90°.

Rhombus All sides are equal in length.Opposite angles are equal in size.

Square All sides are equal in length.All angles are the same and

equal to 90°.

Quadrilaterals

Rectangles

Rhombuses

Squares


Other quadrilaterals Shape Properties

Trapezium One pair of opposite sides is parallel.

Kite Two pairs of adjacent (next to each other) sides are equal in length.

One pair of opposite angles (the ones that are between the sides of unequal length) are equal.

Irregular quadrilateral

This shape does not have any special properties.

Name the following quadrilaterals, giving reasons for your answers.a b

THINK WRITE

a The markings on this quadrilateral indicate that all sides are equal in length and all angles equal 90°. Classify the quadrilateral by finding the matching description in the table.

a The given quadrilateral is a square, since all sides are equal and all angles are 90°.

b The arrows on the sides of this quadrilateral indicate that there are 2 pairs of parallel sides. Find the matching description in the table and hence name the quadrilateral.

b The given quadrilateral is a parallelogram, since it has 2 pairs of parallel sides.

7WORKEDExample

remember1. A quadrilateral is a 2-dimensional closed shape with 4 straight sides.2. All quadrilaterals can be divided into 2 major groups: parallelograms and other

quadrilaterals.3. Parallelograms have 2 pairs of parallel sides and include rectangles, squares

and rhombuses.4. Other quadrilaterals include trapeziums, kites and irregular quadrilaterals.

remember


Types of quadrilaterals

1 Name the following quadrilaterals, giving reasons for your answers.a b c

d e f

2

a This quadrilateral is a: A square B rectangle C kiteD rhombus E parallelogram

b This quadrilateral is a: A trapezium B parallelogram C rhombusD irregular quadrilateral E kite

c This quadrilateral is a: A trapezium B squareC irregular quadrilateral D kiteE parallelogram

3 State whether each of the following statements is true or false.a All squares are rectangles.b All squares are rhombuses.c All rectangles are squares.d Any rhombus with at least one right angle is a square.e A rectangle is a parallelogram with at least one angle equal to 90°.f A trapezium with 2 adjacent right angles is a rectangle.g All rhombuses are kites.h A kite could be a parallelogram.

11C

Rhombuses

Squares

WORKEDExample

7

Rectangles

Quadrilaterals



A rectangle is a quadrilateral because:A it has 4 right angles B it has 2 pairs of parallel sidesC its opposite sides are equal in length D it has 4 straight sidesE it has 2 pairs of parallel sides and 4 right angles.

5 Draw 4 equilateral triangles with side lengths 4 cm and cut them out.a Use 2 of these triangles to make a rhombus. Draw your solution.b Use 3 of these triangles to make a trapezium. Draw your solution.c Use all 4 triangles to make a parallelogram. Draw your solution.

6 Copy and cut out the following set of shapes. Arrange the shapes to form a square. Draw your solution.

7 State the types of quadrilaterals that can be seen in each of the following pictures.

a b

8 In your house, find an example of each type of quadrilateral discussed in this section.Write down the type of quadrilateral and where you found it.

9 The picture at right is made up of equilateral triangles. How many rhombuses can you find in the picture? (One rhombus that is made up of 2 triangles is shown.)



Angles in a quadrilateral

As a result of your investigation you should have discovered the rule that is statedbelow.

The sum of the angles in any quadrilateral is 360°.In the quadrilateral at right

a + b + c + d = 360°

This can be easily demonstrated.In quadrilateral ABCD shown above,

the diagonal BD has been drawn. This diagonal divides the quadrilateral into 2 triangles: triangle ABD and triangle BCD.

In triangle ABD: ∠s + ∠t + ∠u = 180°.In triangle BCD: ∠z + ∠x + ∠y = 180°.So in both triangles together ∠s + ∠t + ∠u + ∠z + ∠x + ∠y = 180° + 180°

or ∠s + ∠t + ∠u + ∠x + ∠y + ∠z = 360°. [1]On the other hand, in the quadrilateral ABCD: ∠DAB = ∠s; ∠ABC = ∠t + ∠x;

∠BCD = ∠y and ∠CDA = ∠u + ∠z.

Sum of angles in a quadrilateralYou will need a ruler and a protractor.1. Draw an irregular quadrilateral in your workbook.2. Measure each of the 4 angles using a protractor.3. Find the sum of the 4 angles.4. Record your results for the irregular quadrilateral into the table below.

5. Repeat steps 1–4 for each of the other quadrilaterals in the table.6. Study your results and write any patterns that you have noticed, regarding the

sum of angles in a quadrilateral.

Parallelograms

Kites

Trapeziums

QuadrilateralFirst angle

Second angle

Third angle

Fourth angle

Sum of angles

Irregular quadrilateral

Parallelogram

Trapezium

Kite

Square

Angle sum in a quadrilateral a

b

c

d

B

A

C

Ds

t x

u z

y

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 441And so ∠DAB + ∠ABC + ∠BCD + ∠CDA = ∠s + ∠t + ∠x + ∠y + ∠u + ∠z.

That is, ∠s + ∠t + ∠u + ∠x + ∠y + ∠z = sum of angles in the quadrilateral. [2]

Placing the two results next to each other, we have:

∠s + ∠t + ∠u + ∠x + ∠y + ∠z = 360° [1]∠s + ∠t + ∠u + ∠x + ∠y + ∠z = sum of angles in the quadrilateral. [2]

Comparing [1] and [2] we observe that the left-hand sides of both equations are thesame. Therefore, the right-hand sides of the equations must also be equal and so thesum of angles in a quadrilateral = 360°.

We can use this rule to find missing angles in quadrilaterals, as shown in theexamples that follow.

Find the value of the pronumeral in the diagram at right.

THINK WRITE

The sum of the angles in a quadrilateral is 360°. So express this as an equation.

b + 80° + 75° + 120° = 360°

Simplify by adding 120°, 80° and 75°. b + 275° = 360°Solve to find the value of b. b = 360° − 275°

b = 85°

120º

80º

75º

b

1

23

8WORKEDExample

Find the value of the pronumeral in the following diagram, giving a reason for your answer.

THINK WRITE

According to the markings, the opposite sides of the given quadrilateral are parallel and equal in length. Therefore, this quadrilateral is a parallelogram. In a parallelogram opposite angles are equal. So state the value of the pronumeral.

Opposite angles in a parallelogram are equal in size. Therefore, x = 72°.

72º

x

9WORKEDExample



1 Find the value of the pronumeral in each of the following diagrams.a b c

d e f

g h i

Find the value of the pronumerals in the following diagram.

THINK WRITE

Form an equation by writing the sum of the angles on one side and 360° on the other side of an equals sign.

k + t + 50° + 136° = 360°

The quadrilateral shown in the diagram is a kite. Angle t and angle 136° are the angles between unequal sides and therefore must be equal in size.

t = 136°

Replace t in the equation with 136°. k + 136° + 50° + 136° = 360°Simplify. k + 322° = 360°Solve to find the value of k. k = 360° − 322°

k = 38°

50º

136ºt

k

1

2

345

10WORKEDExample

rememberThe sum of angles in any quadrilateral is equal to 360°.

remember

11D

Angle sum in a quadrilateral


WORKEDExample

8t 42º

42º 138º110º

115º

50ºbt

120º

18º

20º

t

54º

107º 107º

pm

s

127º

12º

32º

250º c 110º

93º

k

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 4432 Find the value of the pronumeral in each of the following diagrams, giving reasons for

your answers.a b

c d

e f

3 Find the value of the pronumerals in each of the following diagrams.a b c

d e f

4

The value of t in the following diagram is:A 360° B 112° C 222°D 138° E 180°

WORKEDExample

9

78º

m 75º

u

132º t

108º

f

63º

p

73º

z

WORKEDExample

1098º

36º

m m

82ºc

d c

64º

t

t

106º

96º

m

p

75º

91º

t

x

115º

m p

n


42º

t


65°

5

The value of r in the following diagram is:A 117° B 63° C 234°D 126° E 57°

6 This photograph shows the roof of a fast food restaurant. Calculate the value of p.

7 Find the size of the obtuse angle in the kite shown atright.

8 Two angles in a parallelogram are 45° and 135°.Find the other 2 angles.

9 Tom measures 2 angles of a kite at 60° and 110°,but forgets which angle is which. Draw 3 differentkites that Tom may have measured, showing the size of all angles in each diagram.

10 Below are sets of 4 angles. For each of the sets decide whether it is possible to con-struct a quadrilateral. Explain your answer.a 25°, 95°, 140°, 100° b 40°, 80°, 99°, 51°

11 Three angles of a quadrilateral are 60°, 70° and 100°.a What is the size of the fourth angle of this quadrilateral?b How many quadrilaterals with this set of angles are possible?c Construct one quadrilateral with the given angle sizes in your book. (The choice of

the length of the sides is yours.)

Constructing quadrilaterals1. (a) Is it possible to construct a quadrilateral with:

(i) 2 obtuse angles? (ii) 3 obtuse angles? (iii) 4 obtuse angles? (b) Explain your answer in each case. If possible, construct one quadrilateral

of each type in your book.(c) Based on your answers to part (a), complete the following sentence: ‘The

maximum possible number of obtuse angles in a quadrilateral is . . .’2. Is it possible to construct a quadrilateral with 4 acute angles? If this is possible,

construct one such quadrilateral in your book. If this is not possible, explain why it is so.

3. Construct quadrilaterals with exactly 2 right angles so that these right angles are:(a) adjacent (that is, next to each other) (b) opposite to each other.Name the shapes that you have constructed.

4. Is it possible to construct a quadrilateral with: (a) exactly 1 right angle? (b) exactly 3 right angles?Give reasons for your answers.

mmultiple choiceultiple choice117º

r

p

119°

GAME

time

Polygons and polyhedra 01

What did DorWhat did Dorothothyy’’s 3 friends ins 3 friends inthe the WizarWizard of Oz want frd of Oz want fromomthe the wizarwizard?d?

75°

A 40°

E

77°

55°131°

81°N

64°

100°

87°82°

BI

125° 150°

A 38°

R36°

58°42°

55°

D

64°A

T

15°

H

48°

27°

O87°

68° 84°

N

42° 72°

40°103° C

37°

G

59°71°

86°

80°

78°A

43°

E

94°

145°20° A

41°

96°

U

61°

86° 28°

A

96°

R

78°

56°R

123°39°

66° 105° 43° 47° 18° 75° 40° 121° 162° 86° 90° 144° 97°22° 35° 199° 15° 91° 130° 65° 60° 206°

The size of theangles represented by letters ineach of the triangles gives the

puzzle answer code.



Design for a front gate

Val and Peter want to replace their front gate with another of the same design. To have this gate made, they need to supply a diagram of it with all measurements and angles shown.

1. There are 4 different shapes formed by the metal bars of the gate. How many different types of triangles are there? Can you name them?

2. How many types of quadrilaterals are there? Name them.

3. Draw a diagram of the gate showing the length measurements and the one angle that is given.

4. Use this angle to calculate all the remaining angles in the diagram.

5. Explain how you were able to achieve this.

Your turn!

Using a ruler and protractor, design a fence that is to be constructed using metal bars. Include different triangles and quadrilaterals to make your design as interesting as possible. Write a short report describing the shapes you have used and important angles which need to be marked on your design to assist in the construction of the fence.

18 cm

60 cm

8 cm

1 m

27°

MQ 7 Chapter 11 Page 446 Thursday, September 13, 2001 3:45 PM


Polygons

A polygon is any closed shape with 3 or more sides, each of which is a straight line.

Naming polygonsPolygons are named according to the number of sides or angles in the shape. (Note thatthe number of sides in any polygon is equal to the number of angles in the polygon.)The table below gives the names of the most common polygons.

Number of sides Name Number of sides Name

3 triangle 9 nonagon

4 quadrilateral 10 decagon

5 pentagon 11 undecagon

6 hexagon 12 dodecagon

7 heptagon 20 icosagon

8 octagon

Which of the following shapes are polygons?a b c

THINK WRITE

a The shown shape is closed and all of its sides are straight lines. So by definition this shape is a polygon.

a The shape is a polygon.

b Although all sides of this shape are straight lines, it is not closed and hence is not a polygon.

b The shape is not a polygon.

c The shape is closed, but one of the sides is not straight. Therefore this shape is not a polygon.

c The shape is not a polygon.

11WORKEDExample


Can you see why this famous building in the USA is called the Pentagon?

Name the following polygons.a b c

THINK WRITE

a Count the number of sides in the polygon.

a Number of sides = 5

Match the number of sides with the corresponding name in the table.

The polygon is a pentagon.

b Repeat steps 1 and 2 as in part a. b Number of sides = 10 The polygon is a decagon.

c Repeat steps 1 and 2 as in part a. c Number of sides = 8 The polygon is an octagon.

1

2

12WORKEDExample

remember1. A polygon is a closed shape with straight sides. 2. Polygons are named according to the number of sides or angles in the shape.

remember


Polygons

1 Which of the following shapes are polygons?

a b

c d

2 Name the following polygons.

a b c

d e f

3 Draw 2 different examples of each of the following polygons.

a hexagon b quadrilateral c nonagon

d pentagon e octagon f triangle

11EWORKEDExample

11Polygons

Regularpolygons

WORKEDExample

12


4 Count the number of sides on a 50 cent piece and name its shape.

5

What shape is each of the following?a

A quadrilateral B hexagonC octagon D pentagonE heptagon

b A horizontal cross-section of a pencilA quadrilateral B hexagon C octagonD pentagon E circle

6 These patterns are made up of different polygons. Can you name them all?

a b

c d

7 Name one place where you can find these polygons in your home or school.a a triangle b a quadrilateral c a hexagon d an octagon


11.2


1 Name the quadrilateral shown, giving a reason for your answer.

For questions 2 to 5, find the missing angle in each of the quadrilaterals.

2 3

4 5

6 A quadrilateral has 3 angles measuring 56°, 102° and 79°. Find the size of the fourthangle.

7 A rhombus has two angles each of size 78°. Find the size of each of the other twoangles.

8 Name the polygon at right.

9 Draw two different examples of an octagon.

10 What name is given to a polygon with 11 sides?

Constructing polygonsTrace this shape, cut it out and cut along the dotted lines to make 3 pieces. Rearrange the pieces to make the following polygons. (Draw the solutions in your workbook). (a) triangle (b) square(c) rectangle (d) trapezium(e) pentagon (f) parallelogram

2

94º 117º

61ºa

h

77º

71º

109º w

65º

39º

t

A.A. van Leeuw van Leeuwenhoek of Holland wasenhoek of Holland wasthe first person to do this in 1the first person to do this in 166776.6.

7 TRAPEZIUM

8 EQUILATERALTRIANGLE

9 OBTUSE-ANGLED,SCALENE TRIANGLE

10 RHOMBUS

11 REGULAR PENTAGON

12 OBTUSE-ANGLED,ISOSCELES TRIANGLE

13 RECTANGLE

14 RIGHT-ANGLED,ISOSCELES TRIANGLE

1 PARALLELOGRAM

2 RIGHT-ANGLED,SCALENE TRIANGLE

3 ACUTE-ANGLED,ISOSCELES TRIANGLE

4 SQUARE

5 REGULAR HEXAGON

6 ACUTE-ANGLED,SCALENE TRIANGLE

Match up the letter in each polygonwith the number beside the correct name

for the shape to find the answer code.

O

E

D

UN

H

C

P

G

S

I M

T

R

1 2 3 4 5 6 7 8 6 9 10 11 12 11 4 1 8

13 4 11 10 4 5 9 2 11 5 2 14 6



Constructing polygons In the preceding exercises there were some questions involving construction of tri-angles and quadrilaterals. In this section we will discuss further ways of constructingpolygons using a ruler, compass and protractor.

A polygon with all sides equal in length and all angles equal in size is called a regular polygon.

A regular polygon can be constructed in a circle as shown in the following workedexample.

Construct a regular nonagon in a circle of radius 5 cm.

THINK WRITE/DRAW

A nonagon has 9 vertices, so we need to mark 9 points on the circumference. Furthermore, since the nonagon is regular, the vertices must be equidistant from each other (that is, evenly spaced along the circumference). There are 360° in a circle, so divide 360° by 9, to find the distance between each point on the circumference.

Number of sides = 9, so there are 360° ÷ 9 = 40° between each point.

Draw a circle of radius 5 cm.Use a protractor to mark off 9 points on the circle at 40° intervals. (These points are to become the vertices of the nonagon.)

Join the points with straight lines to construct a nonagon.

1

2

0

40

90

3

4

13WORKEDExample

remember1. A regular polygon has all sides of equal length and all angles of equal size.2. To construct a regular polygon in a circle, first divide 360° by the number of

sides. This gives you the angle between the vertices. Then mark off the points on the circumference of the circle and join them together with straight lines.

remember


Constructing polygons

1 Construct a square in a circle of radius 5 cm.

2 Construct a regular pentagon in a circle of radius 5 cm.

3 Construct a regular hexagon in a circle of radius 5 cm.

4 Construct a regular octagon in a circle of radius 5 cm.

5 a Follow the instructions below to produce this design.i Draw a square of side 10 cm.ii Find the midpoint (middle) of each side.iii Join the midpoints to form a new square.iv Repeat steps ii and iii for your new square.

b Repeat steps ii and iii to produce smaller and smaller squares. How many smaller squares can you make?

c Colour in your final design.

6 Draw your solutions for each part of this question in your book.

First draw a square with sides of length 5 cm. a With 1 straight line divide the square into:

i 2 equal rectanglesii 2 equal right-angled trianglesiii 2 equal trapeziums.

b With 2 straight lines divide the square into:i 4 equal trianglesii 4 equal squaresiii 3 equal rectanglesiv 4 equal irregular quadrilaterals.

c With 3 straight lines divide the square into:i 4 equal rectanglesii 6 equal rectangles.

7 Copy the following shape and cut it out. Cut out along the dotted lines and rearrange the pieces to form a square. (Draw the solution in your workbook.)

8

Which of the following is a regular quadrilateral?A A rectangle B A parallelogram C A rhombus D A square E All of the above

11F11.3 WORKED

Example

13

Internal angle of a polygon

GAME

time

Polygons and polyhedra 02



Plans and viewsAn object can be viewed from different angles. Architects and draftspersons often drawplans of building sites and various objects when viewed from the front, the side or thetop.

The front view, or front elevation, is what you see if you are standing directly in frontof an object.

The side view, or side elevation, is what you see if you are standing directly to oneside of the object. You can draw the left view or the right view of an object.

The top view, or bird’s eye view, is what you see if you are hovering directly over thetop of the object looking down on it.

The following object is made from 4 cubes.Draw plans of it showing:a the front view b the right view c the top view.

THINK DRAW

a Make this shape using cubes. Place the shape at a considerable distance and look at it from the front (this way you can see only the front face of each cube). Draw what you see. (Or simply imagine looking at the shape from the front and draw what you see.)

a

b Look at your model from the right, or imagine that you can see only the right face of each cube and draw what you see.

b

c Look at your model from the top, or imagine that you can see only the top face of each cube. Draw what you see.

c

Front

Front view

Right view

Top view

Front

14WORKEDExample


Draw: i the front view ii the right viewiii the top view of this solid.

THINK DRAW

Find an object of similar shape, or visualise the object in your head.Whether viewed from the front, or from the right of the object, the cylindrical shaft will appear as a long thin rectangle. The circular discs will also be seen as a pair of identical rectangles. So the front view and the right view are the same.When the object is viewed from above, all we can see is the flat surface of the top disc; that is, a large and a small circle with the same centre. (Note that such circles are called concentric.)

1

Front view

Right view

Top view

2

3

15WORKEDExample

The front, right and top views of a solid are shown. Use cubes to construct the solid.

THINK CONSTRUCT

Use cubes to construct the solid. Check carefully that your solid matches each of the 3 views you are given. Make adjustments if necessary.

Frontview

Rightview

Topview

Front

1

Front

2

16WORKEDExample


Plans and views

1 The following objects are made from cubes. For every object draw the plans, showingthe front view, the right view and the top view. (You may wish to use a set of cubes orbuilding blocks to help you.)a b c

d e f

g h

remember1. The front view, or front elevation, is what you see if you are standing directly in

front of an object.

2. The side view, or side elevation, is what you see if you are standing directly to one side of the object. You can draw the left view or the right view of an object.

3. The top view, or bird’s eye view, is what you see if you are hovering directly over the top of the object looking down on it.

remember

11GWORKEDExample

14

FrontFront

Front

FrontFront

Front

Front

Front


i j

2 Draw the front, right and top views of each solid shown.

a b c d

3 The front, right and top views of a solid are shown. In each case, use cubes to constructthe solid.

a b

c d

FrontFront

WORKEDExample

15

WORKEDExample

16

Front view

Right view

Top view

Front

Front view

Right view

Top view

Front

Front view

Right view

Top view

Front

Front view

Right view

Top view

Front


The front, right and top views of a solid are shown. Which of the given drawings couldrepresent the solid?

A B C

D E

5 a What shape is the top view of a telephone pole?

b What shape is the top view of the Melbourne Cricket Ground?

c What shape is the side view of a bucket?

d What shape is the top view of a car?

6 a Draw the side view of a pool table.

b Draw the front view of your house (seen from the street).

c Draw the side view of a kettle.

d Draw the top view of your television set.

7 A shape is made using only 4 cubes. Its front view, right view and top view are shown.

a Is it possible to construct this solid?

b Describe or draw what this solid would look like.


Front view Right view Top view

Front

FrontFront

Front

Front Front

Front view Right view Top view

Front


Polyhedra, nets and Euler’s ruleA polyhedron is a 3-dimensional shape in which each flat surface is a polygon.

This photograph of an ancient Egyptian pyramid is an example of a polyhedron, sinceeach of its 5 flat surfaces is a polygon.

The flat surfaces which make a polyhedron are called faces.The lines where 2 faces of a polyhedron meet are called edges.The points where 3 or more edges of a polyhedron meet are called vertices.

Edge

Face

Vertex

For the polyhedron shown, write down:a the number of faces and the shape of each faceb the number of edges c the number of vertices.

THINK WRITE

a The base of the polyhedron shown is a square and the other 4 faces are triangles.

a Number of faces = 5Shape of the faces: 1 square and 4 triangles

b Four edges are formed where each of the triangles meets the square base. Another 4 edges are formed where the triangular faces meet each other.

b Number of edges = 8

c There are 4 vertices on the square base and 1 at the top.

c Number of vertices = 5

17WORKEDExample


Naming polyhedraThe table below gives the names of some common polyhedra that you might often see.The names are often associated with the number of faces of the polyhedron.

PolyhedronNumber of

faces Name

4 tetrahedron, or triangular-based pyramid

5 square-based pyramid

6 cube

8 octahedron

12 dodecahedron

20 icosahedron


Technology and polyhedraA demonstration version of the program Poly is available on the Maths Quest 7CD-ROM. This program allows you to visualise polyhedra and their nets.

When you first open Poly, follow these steps to select the most appropriate options:Go to View then select Available modes. ‘Tick’ the following options:

Option 2: 3-dimensional shaded polyhedra

Option 4: 3-dimensional edges (wireframe)

Option 5: 3-dimensional vertices

Option 6: 2-dimensional net

Poly can be used to assist you in counting the number of faces, edges and vertices aswell as view the shape of each face.

For the polyhedron in worked example 17, follow these steps:1. Select Johnson Solids and Square Pyramid (J1)

2. Press the icon (3-dimensional shaded polyhedra) and rotate the object to

count the number of faces and to see the shape of each face. (You can rotate the solidby placing your mouse arrow over the solid then clicking and holding down themouse while moving the arrow.)

3. Press the icon (wireframe). Rotate until all edges are clearly seen and can be

counted.

4. Press the icon (3-dimensional vertices). Rotate until all vertices are clearly

seen and can be counted.Use Poly to obtain different views of the polyhedra shown in the table by rotating

each solid using option (3-dimensional shaded polyhedra).

Poly


Nets of polyhedraA 2-dimensional plan that can be folded to construct a 3-dimensional polyhedron is called a net of that polyhedron.

This is a net of a square-based pyramid: Square-based pyramidif you fold up the triangles and stick them together,the square-based pyramid shown at right will be formed.

Use Poly to see the nets of different solids by selecting option (2-dimensional

net). You can also see how the solid ‘unfolds’ into a net and then folds back into a poly-hedron by moving the button forward and backwards along the horizontal slot when

using option (3-dimensional shaded polyhedra).

Polyhedra, nets and Euler’s rule

You may wish to use the program Poly to assist you in completing this exercise.1 i ii iii

For each polyhedron shown, write down:a the number of faces and the shape of each faceb the number of edges c the number of vertices.

Poly

remember1. A polyhedron is a 3-dimensional shape in which every flat surface is a polygon.2. The flat surfaces that form a polyhedron are called faces.3. The lines where 2 faces of the polyhedron meet are called edges.4. The points where 3 or more edges of the polyhedron meet are called vertices.5. The net is a 2-dimensional plan, which can be folded to form a 3-dimensional

object.

remember

11HPolyWORKED

Example

17


2 Name each of the polyhedra in question 1.

3 Copy this net of a tetrahedron. Tabs have been included to assist in the construction.Cut out the net and fold it to construct the tetrahedron. Look at the tetrahedron and write down:a the number of facesb the number of vertices c the number of edges.

4 This is the net of a parallelepiped. Each of its faces is a parallelogram.Copy the net, cut it out and fold it to construct the parallelepiped. Look at the parallelepiped and write down:a the number of facesb the number of vertices c the number of edges.

5 This is the net of a truncated tetrahedron. It is a tetrahedron with its corners cut off.Copy the net, cut it out and fold it to construct the truncated tetrahedron. Look at the truncated tetrahedron and write down:a the number of facesb the number of vertices c the number of edges.

6 Copy this net of an icosahedron.

Cut out the net and fold it to construct the icosahedron. Look at the icosahedron andwrite down:a the number of faces b the number of vertices c the number of edges.


465

7 a

Copy this table into your workbook. Use the answers to questions

1

,

3

,

4

,

5

,

6

(orthe program Poly) to complete the table.

b

State the pattern that you have found.

c

The pattern that you have discovered is known as Euler’s rule. (Euler is pro-nounced ‘Oiler’.) Copy the rule into your workbook for future reference.

Euler’s rule for polyhedra can be stated as:for any polyhedron, F

+

V

−

E

=

2,where F is the number of faces, V is the number of vertices and E is the number of edges of the polyhedron.

In other words, the number of faces plus the number of vertices minus the number ofedges equals 2.

8

Use Euler’s rule, established in the previous question, to check whether it is possiblefor a polyhedron to have:

a

6 faces, 8 vertices and 10 edges

b

9 faces, 12 vertices and 19 edges.

9

Use Euler’s rule to answer each of the following questions.

a

A polyhedron has 10 faces and 16 vertices. How many edges does it have?

b


c

A polyhedron has 10 edges and 6 vertices. How many faces does it have?

d


Question number Name of polyhedron

Number of faces

(F)

Number of vertices

(V)

Number of edges

(E) F

+

V

−

E

1 i

cube

1 ii

octahedron

1 iii

dodecahedron

3

tetrahedron

4

parallelepiped

5

truncated tetrahedron

6

icosahedron

UsingEuler’s

rule

MQ 7 Chapter 11 Page 465 Wednesday, June 18, 2003 4:21 PM


10 a Use straws and plasticine (or any other suitable material) to construct these poly-hedra:

i a square-based pyramid ii a nonahedron

b For each model count the number of faces, edges and vertices and hence verifyEuler’s rule.

Making models of polyhedraUse the program Poly to find the nets of 3 polyhedra. There are some suggestions below. Print out each net and trace onto coloured paper or card. (You may like to enlarge your net by using a photocopier first.) Cut and fold to form each of these polyhedra. Have fun!• Hebesphenomegacorona (J89) (in Johnson Solids)• Decagonal deltohedron (in Dipyramids and Deltohedrons)• Hexakis octahedron (in Catalan Solids)• Square anti-prism (in Prisms and Anti-prisms)• Great rhombicosidodecahedron (in Archimedean Solids)

11.3

MA

TH

SQUEST

C H A L L

EN

GE

MA

TH

SQUEST

C H A L L

EN

GE

1 How many rectangles can you find in this shape?

2 Show how to cut this rectangle into 2 pieces that fit together to form asquare?

Poly


Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1 An obtuse-angled triangle has one angle 90°. 2 An triangle has two sides equal in length and two angles

equal in size.3 An equilateral triangle has all sides in length and all angles

equal in size.4 An triangle has all angles less than 90°.5 A triangle has no equal sides or angles. 6 A right-angled triangle has one angle equal to 90° (that is, a angle).7 A kite has two pairs of sides, equal in length. The angles

between sides of a kite are equal in size.8 A has two pairs of opposite sides equal in length and four 90° angles. 9 A square has four and four 90° angles.

10 A has one pair of parallel sides.11 A rhombus has four equal sides. Opposite angles of a rhombus are .12 A parallelogram has two pairs of sides. Opposite sides of a

parallelogram are equal in length and angles are equal in size.13 A polygon is any 2-dimensional shape with edges.14 A polygon has all sides equal in length and all angles equal in size.15 A polyhedron is a object. Each face of a polyhedron is a .16 The sum of the angles in any triangle is equal to degrees.17 An angle of a triangle and an interior angle adjacent to it are .18 The sum of the angles in any quadrilateral is equal to degrees.19 A plan which can be cut out and folded to make a is called

a of that shape.20 The bird’s eye view of an object is its view. 21 The view is drawn when standing directly in front of an object.22 The is the left, or the right view of an object.23 Euler’s rule for a polyhedron states that the number of plus the

number of minus the number of equals 2.

summary

W O R D L I S Tfacesadjacentpolygonclosednetsupplementaryacute-angledrectangle

rightoppositeverticespolyhedrongreater than180straighttop

edgesside elevationunequalequal sidesequal in sizeequal3603-dimensional

frontregularexteriorisoscelesscaleneparalleltrapezium


1 Name the following triangles according to the length of their sides.a b c

2 Name the following triangles according to the size of their angles.a b c

3 Find the value of the pronumeral in each of the following.a b c


5 The Indian teepee shown has an angle of 46° at its peak. What angle does the wall make with the floor?

CHAPTERreview

11A

11A

11Bt

48º 65ºb

62º

x 65º

40º

11Bp

62º

m

52º

n

nn

11B

w

46º

C h a p t e r 1 1 P o l y g o n s a n d p o l y h e d r a 4696 Name the following quadrilaterals, giving reasons for your answers.

a b c

d e f



9 A circus trapeze attached to a rope is shown. Find the size of angle t.

10 Name the following polygons.

a b c

11 Draw two different examples of a hexagon.

11C

11D85º

120º

80ºx

110º

105º

e

36º

42º

240º n

11D110º

74ºw68º

g g

k 126°

11D65º65º

t t

11E

11E


12 Draw a circle of radius 5 cm. Mark points on its circumference every 120°. Join the dots together to construct a triangle.

13 Draw a circle of radius 5 cm. a To construct a hexagon you need to mark 6 points on the circle’s circumference.

Calculate the number of degrees between each point.b Mark off 6 points on the circle and construct a hexagon.

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

15 The front, side and top view of a solid are shown. Construct this solid, using blocks. a b

16 For each polyhedron shown write down:i the number of facesii the number of verticesiii the number of edges.

a b

17 a A polyhedron has 10 faces and 8 vertices. How many edges does it have?b A polyhedron has 6 faces and 12 edges. How many vertices does it have?c Is it possible to have a polyhedron with 10 faces, 6 edges and 4 vertices? Give reasons

for your answer.

11F

11F

11G

Front

Front

11GFront view

Right view

Top view

Front

Front view

Right view

Top view

Front

11H

11Htesttest

CHAPTERyyourselfourself

testyyourselfourself

11

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Polygons and polyhedra - WordPress.com · Polygons and polyhedra. 422 Maths Quest 7 for Victoria Introduction The world around you is ﬁlled with many different shapes and objects