MEASUREMENT, SPACE AND GEOMETRY 7web2.hunterspt-h.schools.nsw.edu.au/studentshared...circumference radius sector tangent arc quadrant segment diameter chord a the distance from the

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MEASUREMENT, SPACE AND GEOMETRY

7 Life would be quite different without circles or circular shapes. The wheel is considered to be one of the greatest inventions in human history. In this chapter, we will study the perimeter and area of circular shapes, and be introduced to a special number called ‘pi’, which has the Greek symbol π.

07 CO NCM8_4 SB TXT.indd Ch07:19407 CO NCM8_4 SB TXT.indd Ch07:194 19/9/08 1:48:54 PM19/9/08 1:48:54 PM

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In this chapter you will:

• identify and name parts of a circle and the related lines: centre, radius, diameter, circumference, arc, sector, quadrant, semi-circle, chord, segment and tangent

• investigate the line symmetry and rotational symmetry of circles, and of diagrams involving circles, such as a sector and a circle with a chord or tangent

• demonstrate by practical means that the ratio of the circumference to the diameter of a circle is constant, and defi ne the number π

• develop formulas to calculate the circumference of circles in terms of the radius, r, or diameter, d: C = πd or C = 2πr

• develop the formula to calculate the areas of circles: A = πr2

• fi nd the area and perimeter of quadrants and semi-circles

• (Stage 5) calculate the perimeter and area of sectors

Wordbank

• annulus the region between two different-sized circles with the same centre; a donut shape

• circumference the distance around a circle; a circle’s outer boundary or perimeter

• diameter the distance from one side of a circle to the other side, through the circle’s centre

• irrational a number that cannot be expressed as an exact fraction or decimal

• pi (π) a special irrational number, approximately 3.1416, used in the calculations of circular measurements

• quadrant a quarter of a circle• radius the distance from the centre of a circle

to the circle’s edge

07 CO NCM8_4 SB TXT.indd Ch07:19507 CO NCM8_4 SB TXT.indd Ch07:195 19/9/08 1:49:09 PM19/9/08 1:49:09 PM

196 NEW CENTURY MATHS 8 STAGE 4 ISBN: 9780170136952

Start upWorksheet7-01

Brainstarters 7

1 Find the perimeter of each of these shapes:

a b c

d e f

2 Find the area of each of the following shapes:

a b c

d e f

7-01 Parts of a circleThe circle is a completely round shape. It can be drawn using compasses. Every point on a circle is exactly the same distance from the circle’s centre.

3 cm

3 cm

3 cm

3 cm

2.4 m

14 m

10 m

29 m

11 m

15 mm

15 m

m

34 mm

20 mm7 cm

2 cm

5 cm

3 cm

Skillsheet7-01

What is area? 1 2 m

2 m

2 m

2 m

10 cm6

cm

7 m

16 m

12 cm10.4

cm

8.2 cm 3 mm 3 mm

7 mm

7 mm

5 m

15 m

5 m 5 m

5 m6 m

Worksheet7-02

Parts of a circle

07 NCM8_4 SB TXT.fm Page 196 Friday, September 19, 2008 2:16 PM

197ISBN: 9780170136952 CHAPTER 7 THE CIRCLE

The circle in the diagram on the right has centre O, and any point on the circle, for example P or Q, is exactly 2 cm from O. This distance, from the centre to the edge of the circle, is called the radius of the circle (plural: radii).

A line running from one side of a circle to the other side and through its centre is called the diameter of the circle.

A circle can be folded in half in an infinite number of ways, along any diameter, so it has an infinite number of axes of symmetry.

A circle can also be rotated through any angle size (in degrees) and still map onto itself, so it has an infinite order of rotational symmetry. The centre of symmetry is the centre of the circle.

The perimeter of a circle is called its circumference. The other parts of a circle and its related lines are shown in the diagrams below.

1 Use a ruler and compasses to construct a circle of:a radius 4 cm b radius 2.5 cm c diameter 6 cm

2 An ellipse is a ‘flattened’ circle.a How many axes of symmetry has an ellipse?b What order of rotational symmetry has an ellipse?

Exercise 7-01

2 cm

Oradius2 cm

P

Q

diameter

centre ofsymmetry

circumference chord

arc

tangent

semi-circle

segment

sectorquadrant



3 Name the parts of the circle marked by letters in these diagrams.

i ii

4 Write a description of each of these in your own words:a diameter b circumference c chordd sector e tangent f segmentg quadrant h arc i radius

5 a Draw and label a sector and a segment in a circle.b What is the difference between a sector and a segment?

6 Which of the following are the parts labelled a and b on this circle? Select A, B, C or D.A diameter and segmentB radius and segmentC diameter and sectorD radius and sector

7 Copy each shape below. Then state: i the number of axes of symmetry in the shape and draw themii the order of rotational symmetry of the shape.

a b

c d

e f

a

c

b da

b

d

c

a

b



8 Match each of the following words to the correct description below:circumference radius sectortangent arc quadrantsegment diameter chord

a the distance from the centre of a circle to its sideb a quarter of a circlec a line that touches the outside of a circle onced the distance from one side of a circle to the other side, through the circle’s centree a line segment from one side of the circle to the other side, not through the centref part of the circumference of a circleg the area inside a circle formed by two radii and an arch the distance around a circlei the area inside a circle formed by a chord and an arc

9 Measure (in millimetres) the diameter and radius of each of the following circles.a b

c

d

10 What is the relationship between the radius and the diameter of a circle?

11 State the mathematical rule connecting d (diameter) and r (radius) as an algebraic formula.

12 Write the diameter of a circle that has a radius of:a 10 cm b 2 cm c 5 mm

13 Write the radius of a circle that has a diameter of:a 10 cm b 2 cm c 1000 mm



WorksheetAppendix 5

2 mm grid paper

Worksheet7-03

Discovering pi

Using technology

Circle partsIn this activity we will use geometry software to construct a circle and display all the circle parts.Step1: Use the Compass Tool to draw a large circle

on the page. An example is shown on the right.

Step 2: Now use the Straightedge Tool, , to draw a radius and a chord in the circle.

Step 3: Use the Text Tool, , to label the radius and chord (as shown above).

1 Construct these circle parts on your page and label each of them:

centre diametercircumference arcsegment sectortangent semi-circlequadrant

The Geometer’s Sketchpad is used in this activity.

Working mathematically

Measuring the circumference of a circleThis is an indoor measuring activity to be done in groups of three.You will need: A measuring tape, or some string and a ruler; graph paper; six round objects such as cans, round cake tins, pipes, money, drums, bottles, and so on.

1 Copy the following results table into your notebook:

Object Diameter, d (mm) Circumference, C (mm) The ratio

a

b

c

d

e

f

Cd----

Applying strategies and reasoning



7-02 What is π?For any circle, = 3.14… This is a special number called pi (pronounced

‘pie’), represented by the Greek letter π. Mathematicians have been trying to find more accurate values for π since the time of the ancient Babylonians and Egyptians. But π has no exact decimal or fraction value, so it is called an irrational number. As a decimal, the digits of π continue endlessly without any repeats or patterns:

π = 3.141 592 653 897 93…The Greek letter π is the first letter of the Greek word perimetron, which means ‘perimeter’ or ‘the measurement around’. This number was named ‘pi’ by the Swiss mathematician Euler in 1737.

Read the information supplied with each question then answer the question, using your calculator.

1 In ancient Egypt (1700 BC) the fraction was the best estimate for pi.

a Change this fraction to a decimal, writing as many decimal places as your calculator provides. (Hint: 256 ÷ 81)

b Round your answer to six decimal places.

Exercise 7-02

Geometry7-01

The value of pi

2 Measure (in millimetres) to find the diameter and the circumference of six different objects. Record your results in the table.

3 Calculate the ratio for each object. Round your answers to two decimal places.

4 Draw a graph similar to the one on the right, showing the diameter and the circumference for each object.

5 What do you notice about the graph?

6 It seems that there is a formula for finding the circumference of a circle. Complete: C = d ×

Cd----

100

200

300

400

500

0

600

700

800

900

1000

Cir

cum

fere

nce

(mm

)

Diameter (mm)50 100 150 200 250 300

circumferencediameter

------------------------------------

25681

---------



2 In ancient Greece, Archimedes (287 BC to 212 BC) claimed that pi was between 3 and 3

a Change these two fractions to decimals, writing as many decimal places as your calculator provides.

b Round your two answers to six decimal places.

3 In the Middle Ages, Leonardo of Pisa (AD 1180 to 1250) claimed that pi was between and

a Change these two fractions to decimals, writing as many decimal places as your calculator provides.

b Round your two answers to six decimal places.

4 In China in AD 400, Tsu Ch’ung-chih, an expert in mechanics, with an interest inmachinery, gave his estimate of pi as

Change this fraction to a decimal.

5 In India the Hindu mathematician Bhaskara claimed that pi was Change this fraction to a decimal.

6 In 2002, Yasumasu Kanada of Japan used a supercomputer to calculate pi to over one trillion decimal places! Here are the first 120 decimal places:3.14159 26535 89793 23846 26433 83279 50288 4197169399 37510 58209 74944 59230 78164 06286 2089986280 34825 34211 70679 82148 08651 32823 06647

The most remarkable thing about pi is that there seems to be no pattern in the numbers (that is, the digits never repeat). We call such a number an irrational number. a Round the value for pi given above to:

i 10 decimal placesii six decimal places

1070------

1071------ .

14404584

9--

------------ 14404581

5--

------------ .

355113--------- .

39271250------------ .



b Copy and complete the following table so that you can compare the decimal values for pi you calculated in Questions 1 to 5.

c In pre-calculator days, was used as an approximation for π because it was more convenient to work with fractions in this form. Compare as a decimal with the computer result for π, and state the number of decimal places to which it is accurate.

d Which of the ancient mathematicians had the most accurate value for pi? Compare that mathematician’s result with the computer result, and state the number of decimal places to which it was accurate.

Place or era Origin Decimal value

Today Computer 3 1 4 1 5 9 2 6 5 3 5

Ancient Egypt

Ancient Greece3

3

Middle Ages

China

India

•

25681

---------

1071------

1070------

14404584

9---

------------

14404581

5---

------------

355113---------

39271250------------

227

------227

------

Mental skills 7

Multiplying or dividing by a multiple of 101 Examine these examples:

a 4 × 700 = 4 × 7 × 100 = 28 × 100 = 2800b 5 × 60 = 5 × 6 × 10 = 30 × 10 = 300c 12 × 40 = 12 × 4 × 10 = 48 × 10 = 480d 3.2 × 30 = 3.2 × 3 × 10 = 9.6 × 10 = 96 (by estimation, 3 × 30 = 90 ≈ 96)e 4.5 × 50 = 4.5 × 5 × 10 = 22.5 × 10 = 225 (by estimation, 5 × 50 = 250 ≈ 225)f 9.4 × 200 = 9.4 × 2 × 100 = 18.8 × 100 = 1880 (by estimation, 9 × 200 = 1800

≈ 1880)2 Now find these products:a 8 × 2000 b 3 × 70 c 11 × 900 d 2 × 300e 4 × 4000 f 5 × 80 g 7 × 70 h 1.3 × 40i 2.5 × 600 j 6.8 × 200 k 3.9 × 50 l 4.4 × 4000

Maths without calculators



3 Examine these examples:a 8000 ÷ 400 = 8000 ÷ 100 ÷ 4 = 80 ÷ 4 = 20b 200 ÷ 50 = 200 ÷ 10 ÷ 5 = 20 ÷ 5 = 4c 6000 ÷ 20 = 6000 ÷ 10 ÷ 2 = 600 ÷ 2 = 300d 282 ÷ 30 = 282 ÷ 10 ÷ 3 = 28.2 ÷ 3 = 9.4e 3520 ÷ 40 = 3520 ÷ 10 ÷ 4 = 352 ÷ 4 = 88f 8940 ÷ 200 = 8940 ÷ 100 ÷ 2 = 89.4 ÷ 2 = 44.7

4 Now find these quotients:a 560 ÷ 70 b 2500 ÷ 50 c 3200 ÷ 400d 440 ÷ 20 e 160 ÷ 40 f 1500 ÷ 30g 450 ÷ 50 h 744 ÷ 80 i 2550 ÷ 300j 846 ÷ 200 k 576 ÷ 60 l 2040 ÷ 50

Just for the record

π = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820…

Ancient civilisations knew about the value of π, estimating it as 3. Over time, the calculations have improved due to better formulas and technology. Since the first computer, the ENIAC, was invented in 1949, much progress has been made. In 2002, a Japanese supercomputer took 602 hours to calculate π to over one trillion decimal places.

In 2006, Japanese counsellor Akira Haraguchi memorised π to 100 000 decimal places. It took him over 16 hours to recite it.On average, how many digits would Haraguchi have recited per minute?

Year Person/Country Number of decimal places

1855 Shanks, England 527

1949 ENIAC computer, USA 2037

1973 CDC 7600 computer, France 1 000 000

1988 Kanada, Hitachi S-820 computer, Japan 200 000 000

1989 Chudnovsky brothers, USA 1 000 000 000

1999 Kanada and Takahashi, Hitachi SR8000 computer, Japan 206 158 430 000

2002 Kanada, Hitachi SR8000 computer, Japan 1 241 100 000 000

??? ??? ???

07 NCM8_4 SB TXT.fm Page 204 Monday, September 29, 2008 1:29 AM


7-03 The circumference of a circleThe circumference of a circle is calculated by multiplying its diameter by the value of π (pi).Pi is often estimated as 3.14, but a more accurate value can be found on your

calculator when you press the key (you may need to press first).• The circumference, C, of a circle with diameter d is C = π × diameter = πd.• Because the diameter of a circle is double its radius, the circumference, C, of a circle with

radius r is C = π × 2 × radius = π2r = 2πr.

Worksheet7-04

A page of circles

Worksheet7-05

Circumferenceand area

π SHIFT

The circumference of a circle is:C = πd (d = diameter) or C = 2πr (r = radius) !

Example 1

a Estimate the circumference of a circle with a diameter of 5 cm.b Calculate the circumference of the circle, correct to two decimal

places.

Solutiona C = πd

= π × 5≈ 3 × 5= 15 cm

b C = πd= π × 5 Press 5 on your calculator.= 15.707 963…≈ 15.71 cm

5 cm

π is approximately 3.

π × =

Worksheet7-06

Circumferenceproblems

Example 2

a Estimate the circumference of a circle with radius 4 cm.b Calculate, correct to one decimal place, the circumference of the

circle.

Solutiona C = 2πr b C = 2πr

= 2 × π × 4 = 2 × π × 4 On your calculator enter:≈ 2 × 3 × 4 = 25.132 74… 2 4 = 24 cm ≈ 25.1 cm

4 cm

× π × =



1 Use π ≈ 3 to approximate the circumference of a circle with:a a diameter of 10 cm b a diameter of 100 cmc a diameter of 2 cm d a diameter of 7 cme a radius of 10 cm f a radius of 3 cmg a radius of 5 cm h a radius of 1 cm

2 Calculate, correct to two decimal places, the circumference of each circle below.

a b c d

3 Calculate the circumference of each circle below, correct to two decimal places.

a b c d

Exercise 7-03

Example 3

Find, correct to two decimal places, the perimeter of:

a this quadrant b this semi-circle

Solution

a Perimeter = × circumference + radius + radius

= × 2 × π × 8 + 8 + 8 C = 2πr

= 28.566 370…≈ 28.57 cm

b Perimeter = × circumference + diameter

= × π × 14 + 14 C = πd

= 35.991 148…≈ 35.99 cm

8 m

14 cm

of circumference

14

8 m

14---

14---

of circumference12

14 cm

12---

12---

Ex 1

d = 4 cm d = 8 m d = 3 cm d = 6 cm

Ex 2

r = 6 m r = 10 mm r = 7 cm r = 2 mm



4 Computer software such as The Geometer’s Sketchpad or Cabri Geometry can be used to accurately construct a circle and calculate its perimeter. Use the link provided to see how.

5 A child’s inflatable swimming pool has a diameter of 1.5 m. Find its circumference.

6 Tina’s bicycle has wheels with a diameter of 60 cm.a How far does the bicycle move when a wheel turns around once?b If Tina cycles 900 m to school, how many complete turns does the bicycle wheel

rotate?

7 This tin of tomatoes has a diameter of 75 mm. If the label wraps around the tin completely, how long is the label? Answer correct to the nearest millimetre.

8 The Earth has a radius of 6370 km. Find the distance around the Equator.

9 A 20-cent coin has a radius of 16 mm. Calculate its circumference.

10 Which of the following intervals best shows the circumference of this circle? Select A, B, C or D.ABCD

11 A circle has a circumference of 50.265 cm. Find its diameter, correct to the nearest centimetre.

12 Tape has been placed on all the lines of this indoor hockey pitch. How much tape was used? Answer correct to one decimal place.

13 A circular running track is 400 m long. Find its radius, correct to two decimal places.

14 The Earth takes one year to orbit the Sun. If the Earth is about 149 million kilometres from the Sun, how far does the Earth travel in one year?

Geometry7-02

Calculating perimeters of circles

d = 75 mm

1 cm

44 m

22 m

9 m

9 m



15 Calculate the perimeter of each of these shapes, correct to one decimal place:

a b c

d e f

g h i j

16 Ali and Billy raced each other around this athletic track. Ali ran along the outside perimeter while Billy ran along the inside perimeter. After one lap of the track, who ran the longer distance, and by how much? Answer correct to the nearest metre.

Ex 3

10 cm3 cm

10 cm

40 mm

6 cm

4 cm

20 mm

15 mm7

cm

20 cm

10 cm

10 m

14 m

80 m

Billy

Ali

20 m

4 m




Belt around the EquatorImagine that we wrapped a giant belt tightly around the Earth, along the Equator. This belt would touch the Earth at all points on its circumference, assuming the Earth was a perfect sphere or ball (with no mountains or valleys).

If 1 metre was added to the length of the giant belt, then it would become loose and not touch the Earth any more. There would be a gap between the Earth and the belt. How big is this gap?

1 In a group of two to four people, guess whether you could:a slip your hand between the belt and the groundb crawl under the beltc sit under the beltd stand under the belt.

2 If the diameter of the Earth is 12 755 metres, calculate the size of the gap and check whether your answer to Question 1 is correct.

Reasoning


Estimating the area of a circleTo estimate the area of a circle, we could place the circle on a grid of square units and approximate the number of squares inside the circle. We could also count the number of squares in a quadrant and then multiply the answer by 4. (For example, the circle on the right has a radius of 2 units. Counting squares in the shaded quadrant gives approximately 3 squares, so an approximate area for the whole circle is 3 × 4 = 12 square units.)

Applying strategies and reasoning Worksheet7-07

Measuring the areaof a circle

Skillsheet7-01

What is area? 1



1 Copy this table into your book or enter it into a spreadsheet:

2 The diagram on the right shows quadrants of different sizes. For each size, find the radius and, by approximating squares and multiplying by 4, estimate the area of the whole circle that has that radius. Write your answers in the ‘Radius, r’ and ‘Area, A’ columns of the table you copied in Question 1.

3 In the ‘r2’ column of the table, calculate the value of the radius squared. For example: 32 = 3 × 3 = 9.

4 In the column of the table calculate, correct to two decimal places, the answers

when A is divided by r2.

5 As the radius, r, increases, do you think your estimate of the area, A, becomes better or worse?

6 As r increases, is the value of increasing or decreasing?

7 As r increases, do you think the value of is becoming more or less accurate? Explain why.

8 Which is closer to the real value of the area of each circle: A = 3 × r2, or A = 4 × r2?

9 From your table, suggest a good value for in this rule for finding the area of a circle: A = × r2.

Circle Radius, r r 2 Area, A

P

Q

R

S

T

U

Ar 2-----

1 2 3 4 5 6 7 8 9 100

P

Q

R

S

U

T

Radius

Cir

cle

Ar2-----

Ar2-----

Ar2-----




Area by cutting out sectorsYou will need: compasses, a ruler, a protractor and a pair of scissors.

1 Draw a circle of radius 10 cm. Use your protractor to divide the circle into 30° sectors. This should make 12 sectors.

2 Cut out all the sectors as neatly as you can.

3 Make a shape like a rectangle with the 12 parts. Colour the parts to alternate the colours.

4 Use a ruler to measure the base and the height of your ‘rectangle’.

5 Now calculate the area of the ‘rectangle’.

6 Which is closer to the real value of the area of the circle: A = 3r2 or A = 4r2?

7 We will now consider a general rule for the area of a circle. A circle of radius, r, is cut up into many sectors and rearranged into a ‘rectangle’.

Write the formula for the circumference of a circle.

8 Explain why the length of the ‘rectangle’ is πr.

9 What is the formula for the area of the ‘rectangle’?

10 Explain why the area of the circle is πr2.

30°10 cm

r

r

πr

Applying strategies and reasoning



7-04 The area of a circle

1 Using A = πr2 calculate, correct to two decimal places, the area of each of the following circles.

a b c

Exercise 7-04

Worksheet7-04

A page of circles

Worksheet7-05

Circumference and area

The area of a circle is:

A = πr 2 (r = radius)!

Example 4

Calculate, correct to two decimal places, the area of a circle with a radius of 7.2 m.

SolutionA = πr2

= π × 7.22 Press 7.2 on your calculator.= 162.8601…≈ 162.86 m2

7.2 m

π × x2 =

Area is measured in square units, such as m2.

Example 5

Calculate, correct to one decimal place, the area of a circle with a diameter of 10 cm.

Solution

r = × 10 = 5 cm

A = πr2

= π × 52 Press 5 on your calculator.= 78.5398…≈ 78.5 cm2

10 cm

12--- The radius is half of the diameter.

π × x2 =

Ex 4

3 cm

12 mm

0.25 m



d e f

g h

2 Find the area of each of these circles. (Give your answers correct to two decimal places.)a radius 9 cm b radius 16 mm c diameter 0.4 md diameter 0.66 m e radius 0.98 mm f radius 12 000 kmg diameter 40 cm h radius 0.75 m i diameter 150 mm

3 Which of the following is the area of this circle, to one decimal place?Select A, B, C or D.A 14.5 m2 B 66.5 m2

C 7.2 m2 D 16.6 m2

4 A sprinkler on the school playing field sprays water in a circular pattern of radius 13.1 m. Calculate the area being sprayed.

5 Kevlar is a very strong light plastic. What area of it is needed to make a solid bicycle wheel of diameter 685.5 mm? Give your answer in square centimetres, correct to two decimal places.

6 A dinner plate has a radius of 14 cm. Calculate its area, to two decimal places.

7 The circular floor of a fishpond is to be covered in plastic. Find, to one decimal place, the area of plastic needed if the diameter of the pond is 2.8 m.

8 Find the approximate cross-sectional area of an almost circular tree trunk whose diameter is 0.75 m. Give your answer correct to two decimal places.

9 A cushion is made from two circles of material with radius 18 cm. Find the area of material that was needed to make the cushion, correct to one decimal place.

2.5 cm 33 mm4.2 cm

Ex 5

7 cm

10 cm

4.6 m

0.75 m



10 This circle has an area of 452.40 cm2. Find:a its radiusb its diameter

11 The area of a compact disc is approximately 113 cm2. Which of the following is closest to its diameter? Select A, B, C or D.A 12 cm B 13 cmC 12.5 cm D 13.1 cm

12 Computer software, such as The Geometer’s Sketchpad or Cabri Geometry can be used to construct a circle and calculate its area. Use the link provided to go to an activity that shows you how.

7-05 Composite areasA composite shape is made of two or more other shapes. Finding the area of composite figures involves more than one calculation.

r

Geometry7-03

Calculating areas of circles

Worksheet7-09

A page ofcircular shapes

Worksheet7-08

Circle area problems

Example 6

Find the area of this quadrant:

Solution

Area = × area of circle

= × πr2

= × π × 102

= 78.5398…≈ 78.54 mm2 (correct to two decimal places)

10 mm

14---

14---

14---

Area is measured in square units, such as mm2.

Example 7

Find the shaded area of each of these shapes:a b

8 cm

8 cm

4 cm4 cm

15 mm

15 mm



1 Find, correct to one decimal place, the area of each of these shapes:

a b c

d e f

2 Find the shaded area of each of these shapes, correct to two decimal places. (Hint: Where appropriate, combine two semi-circles to make a whole circle.)

a b

Exercise 7-05

Solutiona Area = area of square + area of semi-circle

= s2 + × πr2

= 82 + × π × 42

= 89.1327…≈ 89.1 cm2 (correct to one decimal place)

b Shaded area = area of square − area of circle= s2 − πr2

= 152 − π × 7.52

= 48.2854…≈ 48.3 mm2 (correct to one decimal place)

12---

12---

Radius:

r = × 15 = 7.5 mm12---

Ex 6

4 cm 25 mm

2.6 cm

7 cm

15 mm

120°

0.3 m

Ex 7

6 cm

14 cm

14.3 m

5 m



c d

e f

g h

i

3 a Construct this design for yourself, using compasses.b By taking suitable measurements calculate, correct

to two decimal places, the area of the shaded part of the design.

120 cm

140 cm 2 cm

8 cm

10 cm10 cm

3.2 cm

30 mm

14 mm

20 mm

4 m 3 m3 m

1 m

3 m

3 m

400 m

200 m

300 m



4 Each ring-shape or donut-shape shown below is called an annulus, made from two circles that are concentric (with the same centre) but of different size.

a b

For each annulus, find:i the area of the larger circle

ii the area of the smaller circleiii the (shaded) area of the annulus

5 To water this rectangular field, a gardener uses 10 sprinklers, which cover the blue areas shown on the diagram. The three sprinklers in the middle of the field cover circular areas; the four sprinklers on the sides cover semi-circular areas; and the four sprinklers at the corners cover quadrant areas.a What is the diameter of each circle?b What is the radius of each circle?c What is the total (blue) area watered by the sprinklers? (Answer to two decimal

places.)d What percentage of the lawn is watered? (Answer to one decimal place.)

6 The dimensions of a track are shown.The ends are circular.a Find the combined area of the

(blue) track and the (green) grassed central area.

b Find the area of the grassed central area.

c Find the area of the track.d It is decided to cover the track with synthetic grass which costs $24.25 per square

metre. How much will the track surface cost?

7 Suppose that jam tarts have a diameter of 8.2 cm.a How many can you fit on a baker’s tray measuring 67 cm × 33 cm?b How much wasted space is left on a full tray of tarts?

(You may need to draw a diagram.)

7 m5.5 m

9 m

10 m

12 m

8 m

65 m 55 m

90 m




PizzaOtto the Pizza King needs to increase his prices by 5% but his customers want more pizza for their money. Otto decides to increase the diameter of his large pizzas from 40 cm to 42 cm to offset the 5% price rise.

1 Investigate Otto’s solution by answering the following questions.a What are the areas of the 40 cm pizza and the 42 cm pizza?b How much bigger is the 42 cm pizza?c What percentage increase occurs in going from a 40 cm pizza to a 42 cm pizza?d How does this increase compare with the price rise of 5%?e Is Otto offering a fair deal?f What size pizza must Otto make to offer the same value as was offered by the

40 cm pizza?

2 Otto also makes a 30 cm pizza. He hopes to increase the number of customers by lowering the price of this pizza by 10%. However, he can’t afford to make a loss, so the size will need to be reduced. What should be the diameter of the new pizza?

Reasoning and reflecting

Just for the record

The dartboardBrian Gamlin, a British carpenter, devised the current arrangement of numbers on a dartboard back in 1896. The numbers on a standard dartboard are in the following order, going clockwise, starting at the top: 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12 and 5.

Why this particular order?The numbering is designed to penalise poor shots and to cut down on the incidence of ‘lucky shots’. The numbers are placed to encourage accuracy. The placing of the small numbers on either side of the larger numbers punishes inaccuracy. If we aim for 20, and miss, we will hit 5 or 1. If we aim for 17, and miss, we will hit 3 or 2, and so on.The ‘inner bull’ (black centre) of the dartboard is worth 50 points; the ‘outer bull’ (red ring) surrounding it is worth 25 points; the outer red and green ring scores double points; and the inner red and green ring scores triple points.A standard dartboard has been divided into 20 sectors. Find the angle each sector makes at the centre of the circle.Find the area of each sector if the radius of the playing circle is 17 cm.



7-06 Perimeter and area of a sectorTo find the perimeter and area of a sector, we need to use the fact that a sector is a fraction of a circle and there are 360° at the centre of a circle.

Worksheet7-09

A page of circular shapes

Example 8

1 Find the length of the arc AB on the right.

Solution

Length = × circumference

(where is the fraction at the centre of the circle)

= × 2πr

= × 2 × π × 4

= 4.18879 …= 4.2 cm (correct to one decimal place)

2 Find the perimeter of the sector on the right.

Solution

Perimeter = × circumference + 2 radii

= × 2 × π × 16 + 2 × 16

= 43.1701 …= 43.2 m (correct to one decimal place)

60°4 cm

A

B

60360---------

60360---------

60360---------

60360---------

40°

16 m

40360---------

40360---------

Example 9

Find the area of the sector on the right.

Solution

Area = × area of circle

= × πr2

= × π × 112

= 100.3127 …= 100.3 m (correct to one decimal place)

95° 11 m

95360---------

95360---------

95360---------

Worksheet7-06

Circumferenceproblems

Stage 5



1 Find the length of the arc AB each time, correct to one decimal place.

a b c d

2 Find the perimeter of each sector shown in Question 1, correct to one decimal place.

3 Find the area of each of the shaded sectors, giving your answers to two decimal places:

a b

c d

4 Find the shaded area each time, correct to one decimal place.

a b c

5 A car with a rectangular rear windscreen 160 cm long and 70 cm wide has one large wiper of length 75 cm.The wiper covers the shaded area in the diagram on the right. What percentage of the windscreen area is not cleaned by the wiper?

Exercise 7-06

Ex 8

50°3 cm

A

B88°4.1 m

A

B

27°5.6

mm

AB

129°

13 cm

13 cm A

B

Ex 9

100°

5 m

72°

44 mm

195°

9.9 m65 cm

224°

120°5 cm

45° 45°

10 mm

20 mm

6 m 40°

40°

165°

160 cm

70 cm

75 cm

Stage 5



Power plus

1 Find the perimeter of each of the following shapes.(Give your answers correct to one decimal place.)

a b c

2 Find the shaded area each time, correct to one decimal place.

a b c

d e

3 This diagram shows how the field is marked out for the shot put at an athletics event.a What is the total length of the lines used?

(Answer correct to one decimal place.)b What is the area inside the lines?

(Answer correct to one decimal place.

4 Mathsland has introduced a new 30-centcoin, as shown. The centre of the coin is an equilateral triangle, ABC, of side 2 cm.Arcs of circles of radius 2 cm are centred at A, B and C.a Find the area of the sector marked i.b Assuming that the area of triangle ABC is 1.73 cm2, calculate the shaded area.c Calculate the area of the top face of the 30-cent coin.

6 cm

6 cm

2 cm2 cm

10 c

m

300 m

200 m

400 m4 cm

20 mm

1.4

m

2.7 m

1.8 m4 c

m

90°

6 cm

300°

rR

R = 14 mr = 11 m 5 c

m5 cm

40°2.2 m

2.5 m

2 cm

A

C B C

A

B60°

i

2 cm



Language of mathsannulus arc area centrechord circle circumference compassescomposite concentric diameter irrationalperimeter pi (π) quadrant radius/radiisector segment semi-circle tangent

1 What is the relationship between a circle’s radius and its diameter?

2 What is the relationship between a circle’s radius and its circumference?

3 What is the name given to a ‘slice’ of a circle cut from the centre to the edge?

4 What is meant by the statement ‘π is an irrational number’?

5 Why are there two formulas for the circumference of a circle?

6 What is a tangent to a circle?

Topic overview• What did you learn from this chapter that you did not already know?• What did you find difficult in this chapter?• Copy and complete the summary below about your work on the circle. Add colour to

make your overview more useful. Check it with your teacher when you have finished.

Circ

umfe

ren

ce o

f a

circ

le

• C

= πd or C = 2πr, • π (pi) = 3.141 592 653 5... • Perimeter of sem

i-circles and quadrants

THE CIRCLE

arc chord tangent segment sector quadrant diameter

Parts of a circle

radius( × diameter)1

2

Area of a circle

A = πr2

semi-circles

Composite areas

adding andsubtracting

areas

quadrant

Chapter 7 reviewWorksheet7-10

Circle crossword



Chapter revision1 Name each part of the circles shown.

2 For each shape, state: i the number of axes of symmetryii the order of rotational symmetry

a b c d e f

3 Find the radius of a circle with diameter of:a 2 cm b 4.2 m c 3.6 km

4 The mathematician Al-Khwarizmi (in about AD 800) used 3.1416 as an approximation of π. State the number of decimal places to which Al-Khwarizmi was correct.

5 Calculate correct to one decimal place the circumference of each circle below.

a b c d

6 Calculate the perimeter of each of these shapes, correct to two decimal places:

a b c d

7 Calculate the area of each circle in Question 5.

8 Calculate the area of each shape in Question 6.

9 Find the perimeter and area of each of the following shapes. (Give your answers correct to one decimal place.)

a b c

Exercise 7-01

a

d

c

b

g

e

h

f

Exercise 7-01

Exercise 7-01

Exercise 7-02

Exercise 7-03

2 cm

10 mm14 cm

28 mm

Exercise 7-03

6 cm

12 cm

5 cm

4 cm 9 cm

Exercise 7-04

Exercise 7-05

Exercise 7-06

56°

21.4 m

8.7 m

330°

29 cm 3°

Topic testChapter 7



1 Rate each of the following events as either ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’ or ‘certain’.a The Sun will set today.b You will grow to be 4 m tall.c You will receive a good report in maths.d You will select a red card from a pack of cards.e It will rain tomorrow.f You will watch the news on television tonight.

2 List the sample space for each of the following:a a standard dieb this spinner

3 A standard die is rolled.a What is the probability of rolling an even number?b What is the probability of rolling a number greater than 2?

4 A pack of 20 cards contains 10 blue, 6 purple and 4 pink cards. One card is drawn from the pack at random. Find the probability that this card is:a purple b not purple

5 Study the pattern on the right.a Draw the next two steps in the

pattern.b Copy and complete the table below.

R is the number of stars on the bottom row and H is the total number of stars.

c Copy and complete: H = ____ R − ____d Graph the points from the table on a number plane.

6 Copy and complete each of these tables and draw each graph on a number plane.a y = x + 3 b y = 2x c y = 4x − 1

7 Test whether (3, 7) is a point on the line with equation:a y = 5x – 4 b y = 3x – 2

R 2 3 4 5 6

H

Exercise 5-01

Green

Green

Red

Red

Blue

Blue

Yellow

Yellow

Exercise 5-02

Exercise 5-02

Exercise 5-03

1284

Exercise 6-01

Exercise 6-02

x 0 1 2

y

x 1 2 3

y

x -1 0 1

y

Exercise 6-02

Mixed revision 2

07mr2 NCM8_4 SB TXT.fm Page 224 Friday, September 19, 2008 2:37 PM

225ISBN: 9780170136952 MIXED REVISION 2

8 For the graph on the right:a copy and complete the table of values below.b find the equation of the line.

9 a Which two of these lines have the same slope?y = x + 2 y = 2x − 3 y = 2x + 1 y = -2x

b Which two of these lines cross the y-axis at the same point?y = 3x y = x + 3 y = 3x − 1 y = -x + 3

10 a Graph the lines y = x + 3 and y = 2x on the same set of axes.b State the coordinates of the point of intersection of the two lines.

11 Draw a diagram to illustrate each of the following parts of a circle:a a chord b a segment c an arc d a quadrant

12 Find the circumference of each of the circles described below. (Give your answers correct to two decimal places.)a radius 5 cm b radius 12 mm c diameter 20 m d diameter 7 cm

13 Find the circumference of a roller of diameter 0.95 m, correct to two decimal places.

14 Find the perimeter of each of these, correct to two decimal places.

a b c

15 Find the area of each of the circles described in Question 12. (Give your answers correct to two decimal places.)

16 Find the area, correct to two decimal places, of a circular drink coaster whose radius is 7 cm.

17 Find the area of each shape in Question 14. (Answer correct to two decimal places.)

18 Find, correct to two decimal places, the shaded area of each of these shapes:

a b

y

21-2-4 -1-3 43-1

2

1

3

x

Exercise 6-03

x -2 -1 0 1

y

Exercise 6-04

Exercise 6-05

Exercise 7-01

Exercise 7-03

Exercise 7-03

Exercise 7-03

6 cm

20 mm

12 cm

Exercise 7-04

Exercise 7-04

Exercise 7-05

Exercise 7-05

25 mm

3 cm

5 cm

07mr2 NCM8_4 SB TXT.fm Page 225 Friday, September 19, 2008 2:37 PM

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MEASUREMENT, SPACE AND GEOMETRY 7web2.hunterspt-h.schools.nsw.edu.au/studentshared...circumference radius sector tangent arc quadrant segment diameter chord a the distance from the