Chapter 10 Measurement PDF - Year 7 Essential Mathematics

10 Measurement Maths in context: Measurement is important everywhere Measurement skills are essential for all practical The Sydney Harbour Bridge’s surface area is occupations, including: 485 000 m2 (around...

10 Measurement Maths in context: Measurement is important everywhere Measurement skills are essential for all practical The Sydney Harbour Bridge’s surface area is occupations, including: 485 000 m2 (around 60 football fields). all engineers, architects, auto mechanics, bakers, The volume of water in Sydney Harbour is about boiler makers, bricklayers, builders, carpenters 500 gigalitres or five hundred thousand million and chefs litres or 0.5 km3. construction workers, electricians, farmers, The Great Wall of China is more than 6000 km long. hairdressers, house painters, mechanics, miners Australia’s unbroken dingo-proof fence is 5614 km and nurses long and runs from South Australia to Queensland. pharmacists, plumbers, roboticists, seamstresses, The Great Pyramid of Giza, built in 2500 BCE, is sheet metal workers, surveyors, tilers, vets and made of 2.5 million stone blocks that each weigh welders. about 2300 kg (2.3 tonnes), heavier than a Tesla Practical workers use a variety of measurement Model 3 with a mass of 1800 kg or 1.8 tonnes! units, illustrated by these facts. The maximum daytime temperature on Mars is The Eiffel Tower in France is painted with 50 tonnes about 20°C, but a freezing –65°C at night. of paint every 7 years. The Sydney Harbour Bridge is repainted every 5 years by two robots and 100 painters, using 30 000 litres of paint weighing 140 tonnes. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  Chapter contents 10A Metric units of length (CONSOLIDATING) 10B Perimeter (CONSOLIDATING) 10C Circles, π and circumference 10D Arc length and perimeter of sectors and composite shapes (EXTENDING) 10E Units of area and area of rectangles 10F Area of parallelograms 10G Area of triangles 10H Area of composite shapes (EXTENDING) 10I Volume of rectangular prisms 10J Volume of triangular prisms 10K Capacity (CONSOLIDATING) 10L Mass and temperature (CONSOLIDATING) Australian Curriculum 9.0 MEASUREMENT Solve problems involving the area of triangles and parallelograms using established formulas and appropriate units (AC9M7M01) Solve problems involving the volume of right prisms including rectangular and triangular prisms, using established formulas and appropriate units (AC9M7M02) Describe the relationship between π and the features of circles including the circumference, radius and diameter (AC9M7M03) Use mathematical modelling to solve practical problems involving ratios; formulate problems, interpret and communicate solutions in terms of the situation, justifying choices made about the representation (AC9M7M06) ALGEBRA Recognise and use variables to represent everyday formulas algebraically and substitute values into formulas to determine an unknown (AC9M7A01) Solve one-variable linear equations with natural number solutions; verify the solution by substitution (AC9M7A03) Manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables (AC9M7A06) © ACARA Online resources A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto-marked quizzes and much more. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  652 Chapter 10 Measurement 10A Metric units of length CONSOLIDATING LEARNING INTENTIONS To be able to choose a suitable unit for a length within the metric system To be able to convert between metric lengths (km, m, cm and mm) To be able to read a length shown on a ruler or tape measure The metric system was developed in France in the 1790s and is the universally accepted system today. The word metric comes from the Greek word metron, meaning ‘measure’. It is a decimal system where length measures are based on the unit called the metre. The definition of the metre has changed over time. Originally, it was proposed to be the length of a pendulum that beats at a rate of one beat per second. It was later defined as 1/10 000 000 of the distance from the North Pole to the equator on a line on Earth’s surface passing through Paris. In 1960, a metre became 1 650 763.73 wavelengths of the spectrum of the krypton-86 atom in a vacuum. In 1983, the metre was defined as the distance that light travels in 1/299 792 458 seconds inside a vacuum. To avoid the use of very large and very small numbers, an appropriate unit is often chosen to measure a length or distance. It may also be necessary to convert units of length. For example, 150 pieces of timber, each measured in centimetres, may need to be communicated as a total length using metres. Another example might be that 5 millimetres is to be cut from a length of timber A carpenter may need to measure lengths in 1.4 metres long because it is too wide to fit a door opening metres, centimetres and millimetres. Making accurate measurements and converting units are that is 139.5 centimetres wide. essential skills for construction workers. Lesson starter: How good is your estimate? Without measuring, guess the length of your desk, in centimetres. Now use a ruler to find the actual length in centimetres. Convert your answer to millimetres and metres. If you lined up all the class desks end to end, how many desks would be needed to reach 1 kilometre? Explain how you got your answer. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10A Metric units of length 653 KEY IDEAS Metric system × 1000 × 100 × 10 1​ centimetre (cm) = 10 millimetres (mm)​ ​1 metre (m) = 100 centimetres (cm)​ km m cm mm ​1 kilometre (km) = 1000 metres (m)​ ÷ 1000 ÷ 100 ÷ 10 Conversion When converting to a smaller unit, multiply by ​10​or ​100​or ​1000​. The decimal point appears to move to the right. For example: 2.3 m = (2.3 × 100) cm 28 cm = (28 × 10) mm ​ ​  ​ ​ ​    ​ ​ ​ ​ = 230 cm ​ = 280 mm When converting to a larger unit, divide by a power of ​10​(i.e. ​10, 100, 1000​). The decimal point appears to move to the left. For example: 47 mm = (47 ÷ 10) cm 4600 m = (4600 ÷ 1000) km ​ ​  ​ ​ ​    ​ ​ ​ ​ = 4.7 cm ​ = 4.6 km When reading scales, be sure about what units are showing on the scale. This scale shows ​36 mm​. mm 1 2 3 4 cm BUILDING UNDERSTANDING 1 Use the metric system to state how many: a millimetres in ​1​ centimetre b centimetres in ​1​ metre c metres in ​1​ kilometre d millimetres in ​1​ metre e centimetres in ​1​ kilometre f millimetres in ​1​ kilometre. 2 State the missing number or word in these sentences. a When converting from metres to centimetres, you multiply by _______. b When converting from metres to kilometres, you divide by _______. c When converting from centimetres to metres, you _______ by ​100.​ d When converting from kilometres to metres, you ________ by ​1000​. 3 Calculate each of the following. a ​100 × 10​ b 1​ 0 × 100​ c ​100 × 1000​ d ​10 × 100 × 1000​ 4 a When multiplying by a power of 1​ 0​, in which direction does the decimal point move – left or right? b When dividing by a power of 1​ 0​, in which direction does the decimal point move – left or right? Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  654 Chapter 10 Measurement Example 1 Choosing metric lengths Which metric unit would be the most appropriate for measuring these lengths? a dimensions of a large room b thickness of glass in a window SOLUTION EXPLANATION a metres (​ m)​ Using ​mm​or ​cm​would give a very large number, and using km would give a number that is very small. b millimetres ​(mm)​ The thickness of glass is likely to be around ​ 5 mm​. Now you try Which metric unit would be the most appropriate for measuring these lengths? a distance between the centre of the city and a nearby suburb b height of a book on a bookshelf Example 2 Converting metric units of length Convert to the units given in brackets. a ​3 m (cm)​ b ​25 600 cm (km)​ SOLUTION EXPLANATION a 3 m = 3 × 100 cm 1​ m = 100 cm​ ​ ​   ​ ​ ​ = 300 cm Multiply since you are converting to a smaller unit. b 25 ​ =​ 25 600 ÷ 100 000​ There are ​100 cm​in ​1 m​and ​1000 m​in ​1 km​ ​ 600 cm   ​ = 0.256 km and ​100 × 1000 = 100 000​. Now you try Convert to the units given in brackets. a ​2 km (m)​ b ​3400 mm (m)​ Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10A Metric units of length 655 Example 3 Reading length scales Read the scales on these rulers to measure the marked length. a 0 cm 1 cm 2 cm 3 cm b 4m 5m 6m 7m SOLUTION EXPLANATION a 2​ 5 mm​ ​2.5 cm​is also accurate. b ​70 cm​ ​ 1 ​of a metre, which is ​10 cm​. Each division is _ 10 Now you try Read the scales on these rules to measure the marked length. a Answer in millimetres. 0 cm 1 cm 2 cm 3 cm b Answer in centimetres. 0m 1m 2m 3m Exercise 10A FLUENCY 1–3, 4–7​(​ 1​/2​)​ 2, 4–7​(​ 1​/2​)​,​ 8 2, 4–7​(​ 1​/2​)​, 8, 9 Example 1 1 Which metric unit would be the most appropriate for measuring the following? a the distance between two towns b the thickness of a nail c height of a flag pole d length of a garden hose e dimensions of a small desk f distance across a city Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  656 Chapter 10 Measurement Example 1 2 Choose which metric unit would be the most suitable for measuring the real-life length indicated in each of these photos. a b c d e f Example 2a 3 Convert to the units given in the brackets. a ​4 m (cm)​ b ​6 km (m)​ c ​30 mm (cm)​ d ​700 cm (m)​ Example 2a 4 Convert these measurements to the units shown in brackets. a ​5 cm (mm)​ b ​2 m (cm)​ c 3​.5 km (m)​ d 2​ 6.1 m (cm)​ e ​40 mm (cm)​ f ​500 cm (m)​ g ​4200 m (km)​ h ​472 mm (cm)​ i ​6.84 m (cm)​ j ​0.02 km (m)​ k ​9261 mm (cm)​ l ​4230 m (km)​ 5 Add these lengths together and give the result in the units shown in brackets. a ​2 cm​and ​5 mm (cm)​ b ​8 cm​and ​2 mm (mm)​ c 2​ m​and ​50 cm (m)​ d ​7 m​and ​30 cm (cm)​ e ​6 km​and ​200 m (m)​ f ​25 km​and ​732 m (km)​ Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10A Metric units of length 657 Example 2b 6 Convert to the units shown in the brackets. a ​3 m (mm)​ b ​6 km (cm)​ c 2​.4 m (mm)​ d ​0.04 km (cm)​ e ​47 000 cm (km)​ f ​913 000 mm (m)​ g ​216 000 mm (km)​ h ​0.5 mm (m)​ i ​0.7 km (mm)​ Example 3 7 These rulers show centimetres with millimetre divisions. Read the scale to measure the marked length. i in centimetres ii in millimetres a b 0 1 2 0 1 2 3 4 5 c d 0 1 2 0 1 2 3 e f 0 1 2 3 4 0 1 2 3 4 5 g h 0 1 2 3 0 1 2 3 4 8 Read the scale on these tape measures. Be careful with the units given! a b 0 1 2 3 11 12 13 m km 9 Use subtraction to find the difference between the measurements, and give your answer with the units shown in brackets. a ​9 km, 500 m (km)​ b ​3.5 m, 40 cm (cm)​ c ​0.2 m, 10 mm (cm)​ PROBLEM-SOLVING 10–12 12–15 14–17 10 Arrange these measurements from smallest to largest. a ​38 cm, 540 mm, 0.5 m​ b 0​.02 km, 25 m, 160 cm, 2100 mm​ c ​0.003 km, 20 cm, 3.1 m, 142 mm​ d ​0.001 km, 0.1 m, 1000 cm, 10 mm​ 11 Joe widens a ​1.2 m​doorway by ​50 mm​. What is the new width of the doorway, in centimetres? 12 Three construction engineers individually have plans to build the world’s next tallest tower. The Titan tower is to be ​ 1.12 km​tall, the Gigan tower is to be ​109 500 cm​tall and the Bigan tower is to be ​1210 m​tall. Which tower will be the tallest? Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  658 Chapter 10 Measurement 13 Steel chain costs ​$8.20​per metre. How much does it cost to buy chain of the following lengths? a ​1 km​ b ​80 cm​ c ​50 mm​ 14 A house is ​25​metres from a cliff above the sea. The cliff is eroding at a rate of ​40 mm​per year. How many years will pass before the house starts to fall into the sea? 15 Mount Everest is moving with the Indo-Australian plate at a rate of about ​10 cm​per year. How many years will it take to move ​5 km​? 16 A ream of 5​ 00​sheets of paper is ​4 cm​thick. How thick is ​1​sheet of paper, in millimetres? 17 A snail slides ​2 mm​every ​5​seconds. How long will it take to slide ​1 m​? REASONING 18 18 18, 19 18 Copy this chart and fill in the missing information. km m ÷ 100 cm × 10 mm 19 Many tradespeople measure and communicate with millimetres, even for long measurements like timber beams or pipes. Can you explain why this might be the case? ENRICHMENT: Very small and large units – – 20 20 When ​1​metre is divided into ​1​million parts, each part is called a micrometre (​μm​). At the other end of the spectrum, a light year is used to describe large distances in space. a State how many micrometres there are in: i ​1 m​ ii ​1 cm​ iii ​1 mm​ iv ​1 km​ b A virus is ​0.000312 mm​wide. How many micrometres is this? c Research the length called the light year. Explain what it is and give example of distance using light years, such as to the nearest star other than the Sun. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10B Perimeter 659 10B Perimeter CONSOLIDATING LEARNING INTENTIONS To understand that perimeter is the distance around the outside of a two-dimensional shape To understand that marks can indicate two (or more) sides are of equal length To be able to find the perimeter of a shape when the measurements are given The distance around the outside of a two-dimensional shape is called the perimeter. The word perimeter comes from the Greek words peri, meaning ‘around’, and metron, meaning ‘measure’. We associate perimeter with the outside of all sorts of regions and objects, like the length of fencing surrounding a block of land or the length of timber required to frame a picture. This fence marks the perimeter (i.e. the distance around the outside) of a paddock. Farmers use the perimeter length to calculate the number of posts needed for the fence. Lesson starter: Is there enough information? This diagram includes only 90° angles and only one side length is marked. Discuss if there is enough information given in the diagram 10 cm to find the perimeter of the shape. What additional information, if any, is required? KEY IDEAS Perimeter, sometimes denoted as P, is the distance around the 1.6 cm outside of a two-dimensional shape. 2.8 cm Sides with the same markings are of equal length. The unknown lengths of some sides can sometimes be determined 4.1 cm by considering the given lengths of other sides. P = 1.6 + 1.6 + 2.8 + 4.1 = 10.1 cm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  660 Chapter 10 Measurement BUILDING UNDERSTANDING 1 These shapes are drawn on ​1 cm​grids. Give the perimeter of each. a b 2 Use a ruler to measure the lengths of the sides of these shapes, and then find the perimeter. a b c d Example 4 Finding the perimeter Find the perimeter of each of these shapes. a b 3m 3 cm 6m 5m 5 cm 2m SOLUTION EXPLANATION a perimeter = 2 × 5 + 3 There are two equal lengths of ​5 cm​ and ​    ​ ​ ​ ​ = 13 cm one length of ​3 cm​. b perimeter = 2 × 6 + 2 × 8 3m ​    ​ ​ ​ 6−2=4m ​ = 28 m 6m 5m 2m 3+5=8m Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10B Perimeter 661 Now you try Find the perimeter of each of these shapes. a 6 cm b 4m 3 cm 6m Exercise 10B FLUENCY 1–4 2–5 2, 4, 5 Example 4a 1 Find the perimeter of each of the shapes. a b 4 cm 7m 6 cm 11 m Example 4a 2 Find the perimeter of these shapes. (Diagrams are not drawn to scale.) a b 8m 3 cm 5 cm 6m 7 cm 8m 5m 10 m c d 1m 0.2 m 10 km 5 km e f 10 cm 2.5 cm 6 cm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  662 Chapter 10 Measurement Example 4b 3 Find the perimeter of each of these shapes. a 2m b 5m 4 cm 3m 2m 3 cm 4 cm 2 cm Example 4b 4 Find the perimeter of these shapes. All corner angles are ​90°​. a 4 cm b 2m 5m 8m 10 cm 7 cm 5 cm 4m c d 6 km 5 mm 1 mm 3 km 3 mm 9 km 5 a A square has a side length of ​2.1 cm​. Find its perimeter. b A rectangle has a length of ​4.8 m​and a width of ​2.2 m​. Find its perimeter. c An equilateral triangle has all sides the same length. If each side is ​15.5 mm​, find its perimeter. PROBLEM-SOLVING 6, 7 7–9 9–12 6 A grazing paddock is to be fenced on all sides. It is rectangular in shape, with a length of ​242 m​and a width of ​186 m​. If fencing costs ​$25​per metre, find the cost of fencing required. 7 The lines on a grass tennis court are marked with chalk. All the measurements are shown in the diagram and given in feet. 39 feet 21 feet 36 feet 27 feet a Find the total number of feet of chalk required to do all the lines of the given tennis court. b There are ​0.305​metres in ​1​foot. Convert your answer to part a to metres. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10B Perimeter 663 8 Only some side lengths are shown for these shapes. Find the perimeter. (Note: All corner angles are 9​ 0°​.) a 20 mm b 4 cm 10 cm 15 mm 18 cm 9 Find the perimeter of each of these shapes. Give your answers in centimetres. a b 271 mm c mm 430 168 mm 0.38 m 1.04 m 7.1 cm 10 A square paddock has ​100​equally-spaced posts that are ​4​metres apart, including one in each corner. What is the perimeter of the paddock? 11 The perimeter of each shape is given. Find the missing side length for each shape. a 4 cm b c 2 cm ? ? ? P = 11 cm P = 20 m 12 km P = 38 km 12 A rectangle has a perimeter of ​16 cm​. Using only whole numbers for the length and width, how many different rectangles can be drawn? Do not count rotations of the same rectangle as different. REASONING 13 13, 14 14, 15 ​ = 2a + b​) to describe the perimeter of each shape. 13 Write an algebraic rule (e.g. P a a b c a b b a d e f b a a b a c Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  664 Chapter 10 Measurement 14 Write an algebraic rule for the perimeter of each given shape. a a b b a c c b 15 a A square has perimeter P. Write an expression for its side length. b A rectangle has perimeter P and width a. Write an expression for its length. ENRICHMENT: Picture frames – – 16 16 The amount of timber used to frame a picture depends on the outside lengths of the overall frame. Each side is first cut as a rectangular piece, and two corners are then cut at 45° to make the frame. a A square painting of side length 30 cm is to be framed with 30 cm 5 cm timber of width 5 cm. Find the total length of timber required for the job. 30 cm b A rectangular photo with dimensions 50 cm by 30 cm is framed with timber of width 7 cm. Find the total length of timber required to complete the job. c Kimberley uses 2 m of timber of width 5 cm to complete a square picture frame. What is the side length of the picture? d A square piece of embroidery has side length a cm and is framed by timber of width 4 cm. Write an expression for the total length of timber used in cm. The perimeter of a house is found by adding the lengths of all the outside walls. Scaffolding is required around the perimeter for construction work above ground level. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10C Circles, π and circumference 665 10C Circles, π and circumference LEARNING INTENTIONS To know the features of a circle including the radius, diameter and circumference To know that π is the ratio of the circumference of a circle to its diameter To be able to calculate a circle’s circumference, diameter or radius if given one of the other two measurements The special number π, pronounced pi, has fascinated mathematicians, scientists, designers and engineers since the ancient times. This mathematical constant, which connects a circle’s circumference to its diameter, is critical in any calculation that involves circular shapes. It also appears in many other areas of mathematics. Evidence suggests that the Egyptians used the approximation _ 22 to help make calculations 7 and that the Babylonians used the number _25 = 3.125. Around 250 ad, the Greek mathematician 8 Archimedes used an algorithm to prove that π was between 3.1410 and 3.1429 and in a similar manner around 265 ad, a Chinese mathematician, Liu Hui, proved that π ≈ 3.1416. Today we can use computers and complex mathematics to approximate π correct to millions of decimal places. Lesson starter: How close can you get? For this activity, you will need some string, scissors, a ruler and a pair of compasses. We know that π is the C. In this activity, we will estimate the ratio of a circle’s circumference, C, to its diameter, d, and so π = _ d value of π by drawing a circle and measuring both its diameter and circumference. Use a pair of compasses to construct a circle of any size and draw in a diameter. Use a piece of string to trace around the circle and cut to size. Measure the length of the string (the circumference) and the circle’s diameter. Use your measurements to approximate the value of π using π = _ C. d Calculate a class average to see how this approximation compares to an accurate value of π. KEY IDEAS Features of a circle Diameter (d) is the distance across the centre of a circle. cumference Radius (r) is the distance from the centre to the circle. Note: d = 2r. cir Chord: A line interval connecting two points on a circle. eter Tangent: A line that touches the circle at a point. diam – A tangent to a circle is at right angles to the radius. radius Sector: A portion of a circle enclosed by two radii and a portion of a circle (arc). Segment: An area of a circle ‘cut off’ by a chord. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  666 Chapter 10 Measurement Circumference (​ ​C)​ ​is the distance around a circle. t en ​ = 2πr​or ​C = πd​ C g m se hord Pi ​(π)​is a constant numerical value and is an irrational number, c sector meaning that it cannot be expressed as a fraction. – As a decimal, the digits have no pattern and continue forever. The ratio of the circumference to the diameter of any circle is nt C t a nge equal to pi ​(π)​; i.e. ​π = _ ​ ​​. d ​π = 3.14159​(correct to 5​ ​decimal places) – Common approximations include ​3​, ​3.14​and _ ​22 ​​. 7 – A more precise estimate for pi can be found on most calculators or on the internet. BUILDING UNDERSTANDING 1 State whether the following statements relating features of a circle are true or false. a ​r = 2d​ b ​C = 2πr​ c ​d = πC​ C _ d ​π = ​ ​ C _ e ​π = ​r ​ f ​d = 2r​ d 2 Name the features of the circles shown. f d g a e b c 3 The following relate to a circle's radius and diameter. a Find a circle's diameter if its radius is 1​ 5​ mm. b Find a circle's radius if its diameter is 8​ ​ cm. Example 5 Approximating π ​​ A common approximation of ​π​is the fraction _ ​22 ​​. 7 22 _ a Write ​ ​as a decimal rounded to: 7 i two decimal places ii three decimal places. b Decide if _​22 ​is a better approximation to ​π​than the number ​3.14​. 7 Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10C Circles, π and circumference 667 SOLUTION EXPLANATION a i ​22 ​= 3.14 (​ ​2 d.p.​)​ _ ​_22 ​= 3.142​... so round down for ​2​decimal places 7 7 22 ​= 3.143 (​ ​3 d.p.​)​ ii ​_ ​22 ​= 3.1428​... so round up for ​3​decimal places _ 7 7 ​22 ​= 3.14285​... and ​π = 3.14159​... b _ ​22 ​− π = 0.0013​(​ ​4 d.p.​)​and ​π − 3.14 = 0.0016​(​ ​4 d.p.​)​ _ 7 7 So, _​22 ​is a better approximation 22 ​is closer to ​π​. So, ​_ 7 7 compared to ​3.14​. Now you try A possible approximation of π ​25 ​​. ​ ​is the fraction _ 8 a Write ​_25 ​as a decimal rounded to: 8 i two decimal places ii three decimal places. b Decide if _ ​25 ​is a better approximation to ​π​than the number ​3.14​. 8 Example 6 Calculating the circumference Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a b m 7c 2.3 mm SOLUTION EXPLANATION a C = πd Since you are given the diameter, use the ​ ​ ​ =​​ π(7)​ ​ ​ C = πd​formula. Substitute ​d = 7​and use ​ = 21.99 cm a calculator for the value of ​π​. Round as required. Continued on next page Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  668 Chapter 10 Measurement b C = 2πr Since you are given the diameter, use the = 2π(2.3) C = 2πr formula. Substitute r = 2.3 and use = 14.45 mm a calculator for the value of π. Round as required. Now you try Calculate the circumference of these circles using a calculator and rounding your answer correct to two decimal places. a b 4.7 cm 11 mm Exercise 10C FLUENCY 1, 2, 3–5(1/2) 2, 3–5(1/2) 2, 3–5(1/4) Example 5 1 28. A reasonable approximation of π is the fraction _ 9 a Write 28 _ as a decimal rounded to: 9 i two decimal places ii three decimal places. b Decide if _28 is a better approximation to π than 9 the number 3.14. Example 5 355. 2 A good approximation of π is the fraction _ 113 a Write _355 as a decimal rounded to: 113 i three decimal places ii four decimal places. 355 is a better approximation to π than b Decide if _ 113 the number 3.142. The Ancient Greek mathematician and astronomer Archimedes was able to prove that π was between 223 and _ _ 22. 71 7 Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10C Circles, π and circumference 669 Example 6a 3 Calculate the circumference of these circles using ​C = πd​and round your answer correct to two decimal places. a b 13 m cm 5c c d 9.8 m 4.6 m Example 6b 4 Calculate the circumference of these circles using ​C = 2πr​and round your answer correct to two decimal places. a b 4 mm 21 cm c d 16.2 m 8.3 mm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  670 Chapter 10 Measurement 5 Estimate the circumference of these circles using the given approximation of ​π​. a b 2 cm m 5m p =3 p = 3.14 c d 113 14 m cm 22 355 p= p= 7 113 PROBLEM SOLVING 6, 7 6–8 7–9 6 A log in a wood yard has a diameter of 4​ 0 cm​. Find its circumference, correct to one decimal place. ​C 7 We know that the rule ​C = πd​can be rearranged to give ​d = _π ​. Use this rearranged rule to calculate the following, correct to two decimal places. a a circle’s diameter if the circumference is ​10 cm​ b a circle’s diameter if the circumference is ​12.5 m​ c a circle’s radius if the circumference is ​7 mm​ d a circle’s radius if the circumference is ​37.4 m​ 8 A circular pond has radius ​1.4​metres. Find its circumference, correct to one decimal place. 9 A bicycle wheel completes ​1024​ revolutions while on a Sunday ride. Find the distance travelled by the bicycle if the wheel’s diameter is ​85 cm​. Round to the nearest metre. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10C Circles, π and circumference 671 REASONING 10 10, 11 11, 12 10 The following are four fractions which ancient cultures used to approximate ​π​. Egyptian _ ​22 ​ 7 Babylonians _ ​25 ​ 8 Indian _​339 ​ 108 Chinese _​142 ​ 45 a Which one of these ancient historical approximations is closest to the true value of ​π​? Note that ​π​ written to five decimal places is ​3.14159​. ​22 ​has been a very popular approximation throughout many countries. b In more recent times, _ 7 Explain why you think this is the case. c Find your own simple fraction which is a reasonable approximation of ​π​. Write your fraction as a decimal and compare it with ​π​written in decimal form with at least five decimal places. 11 If ​C = πd​, find a rule for the radius of a circle, r​ ​, in terms of its circumference, ​C​, and use it to find the radius of a circle if its circumference is ​25 cm​. Round to two decimal places. 12 A professor expresses the circumference of a circle with diameter ​5 cm​using the exact value, ​5π cm​. Use an exact value to give the circumference of a circle with the given diameter or radius. a Diameter ​7 cm​ b Radius ​3 mm​ ENRICHMENT: Planetary circumnavigation – – 13 13 While not being perfect spheres, the circumference of the planets in our solar system can be approximated by using a circle. Here are some planetary facts with some missing details. Calculate the missing numbers, rounding to the nearest integer. Part Planet Diameter (​km​ approx.) Circumference (​km​ approx.) a Earth ​12 750​ b Mars ​ 6 780​ c Jupiter ​439 260​ d Mercury ​ 15 330​ Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  672 Chapter 10 Measurement 10D Arc length and perimeter of sectors and composite shapes EXTENDING LEARNING INTENTIONS To know that an arc is a portion of a circle To understand how the length of an arc relates to the angle at the centre of the circle To be able to calculate the length of an arc To be able to calculate the perimeter of a sector Whenever a portion of a circle’s circumference is used in a diagram or construction, an arc is formed. To determine the arc’s length, the particular fraction of the circle is calculated by considering the angle at the centre of the circle that defines the arc. Lesson starter: The rule for finding an arc length Complete this table to develop the rule for finding an arc length (l). Angle Fraction of circle Arc length Diagram 180° 180 = _ _ 1 1 × πd l=_ 360 2 2 180° 90° 90 = ________ _ l = __ _ _ _ _ × _ _ _ _ __ 360 90° 45° 30° 150° 𝜃 KEY IDEAS A circular arc is a portion of the circumference of a circle. In the diagram: l r = radius of circle r θ 𝜃 = number of degrees in the angle at the centre of a circle l = arc length r Formula for arc length: l=_ 𝜃 × 2πr or l = _𝜃 × πd 360 360 r r The sector also has two straight edges, each with length of r. θ Formula for perimeter of sector: P=_ 𝜃 × 2πr + 2r or P = _ 𝜃 × πd + d l 360 360 Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10D Arc length and perimeter of sectors and composite shapes 673 Common circle portions quadrant semicircle r r 1 1 P= × 2πr + 2r P= × 2πr + 2r 4 2 A composite figure is made up of more than one basic shape. BUILDING UNDERSTANDING 1 What fraction of a circle is shown in these diagrams? Name each shape. a b 2 What fraction of a circle is shown in these sectors? Simplify your fraction. a b c 60° d e f 315° 120° 3 Name the two basic shapes that make up these composite figures. a b c Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  674 Chapter 10 Measurement Example 7 Finding an arc length Find the length of each of these arcs for the given angles, correct to two decimal places. a b 2 mm 50° 230° 10 cm SOLUTION EXPLANATION a l = _ ​ 50 ​× 2π × 10 The fraction of the full circumference is _​ 50 ​​ ​​  ​ 360 ​ 360 ​ = 8.73 cm and the full circumference is ​2πr​, where ​r = 10​. b l = ​_ 230 ​× 2π × 2 ​230 ​​ The fraction of the full circumference is _ ​​   ​ 360 ​ 360 ​ = 8.03 mm and the full circumference is ​2πr​. Now you try Find the length of each of these arcs for the given angles, correct to two decimal places. a b 7 cm 112° 5m 290° Example 8 Finding the perimeter of a sector Find the perimeter of each of these sectors, correct to one decimal place. a b 5 km 3m SOLUTION EXPLANATION a P =_ ​1 ​× 2π × 3 + 2 × 3 The arc length is one-quarter of the ​ ​  ​4 ​ circumference and included are two radii, each ​ = 10.7 m of ​3 m​. b P =_ ​1 ​× π × 5 + 5 A semicircle’s perimeter consists of half the ​ ​  ​2 ​ circumference of a circle plus a full diameter. ​ = 12.9 km Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10D Arc length and perimeter of sectors and composite shapes 675 Now you try Find the perimeter of each of these sectors, correct to one decimal place. a b 9 mm 1.6 cm Example 9 Finding the perimeter of a composite shape Find the perimeter of the following composite shape, correct to one decimal place. 10 cm 5 cm SOLUTION EXPLANATION ​1 ​× 2π × 5 P = 10 + 5 + 10 + 5 + _ There are two straight sides of ​10 cm​and ​5 cm​ ​ ​    ​ 4 ​ shown in the diagram. The radius of the circle ​ = 37.9 cm is ​5 cm​, so the straight edge at the base of the diagram is ​15 cm​long. The arc is a quarter circle. Now you try Find the perimeter of the following composite shape, correct to one decimal place. 3 cm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  676 Chapter 10 Measurement Exercise 10D FLUENCY 1,​(​ 1​/2)​ ​,​ 2–4 1–4​(​ 1​/2)​ ​ 1​​(1​/3)​ ​,​ 2–4​​(1​/2)​ ​ Example 7 1 Find the length of each of the following arcs for the given angles, correct to two decimal places. a b 60° 8 cm 4m 80° c d 100° 225° 7 mm 0.5 m e f 300° 0.2 km 330° 26 cm Example 8a 2 Find the perimeter of each of these quadrants, correct to one decimal place. a 4 cm b 10 m c d 2.6 m 8 cm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10D Arc length and perimeter of sectors and composite shapes 677 Example 8b 3 Find the perimeter of these semicircles, correct to one decimal place. a b 14 0 mm km 2 c d 4.7 m 17 mm 4 Find the perimeter of these sectors, correct to one decimal place. Include the two radii in each case. a b 60° 4.3 m 115° 3.5 cm c d 6 cm 250° 270° 7m PROBLEM-SOLVING 5, 6 5–7 5–8 Example 9 5 Find the perimeter of each of these composite shapes, correct to one decimal place. a b 2 cm c 3m 30 cm 4m 12 cm d e f 6 km 10 km 10 cm 10 mm 8 cm Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  678 Chapter 10 Measurement 6 A window consists of a rectangular part of height ​2 m​and width ​1 m​, with a semicircular top having a diameter of ​1 m​. Find its perimeter, correct to the nearest ​1 cm​. 7 For these sectors, find only the length of the arc, correct to two decimal places. a b c 85° 140° 1.3 m 100 m 7m 330° 8 Calculate the perimeter of each of these shapes, correct to two decimal places. a b c 5m 4 cm 9m 10 m REASONING 9 9, 10 9–11 9 Give reasons why the circumference of this composite shape can be found by simply using the rule ​ P = 2πr + 4r​. r 10 Explain why the perimeter of this shape is given by ​P = 2πr​. r Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10D Arc length and perimeter of sectors and composite shapes 679 11 Find the radius of each of these sectors for the given arc lengths, correct to one decimal place. a 4m b 60 m c km 45 30° r 100° r r ENRICHMENT: Exact values and perimeters – – 12, 13 12 The working to find the exact perimeter of this composite shape is given by: P = 2×8+4+_ ​1 ​π × 4 ​ ​  ​ 2 ​ ​ = 20 + 8π cm 4 cm 8 cm Find the exact perimeter of each of the following composite shapes. a b 6 cm 2 cm 5 cm 13 Find the exact answers for Question 8 in terms of ​π​. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  680 Chapter 10 Measurement 10E Units of area and area of rectangles LEARNING INTENTIONS To understand what the area of a two-dimensional shape is To be able to convert between metric areas (square millimetres, square centimetres, square metres, square kilometres, hectares) To be able to find the area of squares and other rectangles Area is measured in square units. It is often referred to as the amount of space contained inside a flat (i.e. plane) shape; however, curved three-dimensional (3D) solids also have surface areas. The amount of paint needed to paint a house and the amount of chemical needed to spray a paddock are examples of when area would be considered. Agricultural pilots fly small planes that spray crops with fertiliser or pesticide. The plane is flown up and down the paddock many times, until all the area has been sprayed. Lesson starter: The 12 cm2 rectangle A rectangle has an area of 12 square centimetres (12 cm2). Draw examples of rectangles that have this area, showing the length and width measurements. How many different rectangles with whole number dimensions are possible? How many different rectangles are possible if there is no restriction on the type of numbers allowed to be used for length and width? KEY IDEAS The metric units of area include: 1 mm 1 square millimetre (1 mm2) 1 mm 1 square centimetre (1 cm2) 1 cm2 = 100 mm2 1 cm 1 cm 1 square metre (1 m2) 1 m2 = 10 000 cm2 1 m (Not drawn to scale.) 1m Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10E Units of area and area of rectangles 681 1​ ​square kilometre (​ 1 ​km​2​ ​)​ ​1 ​km​2​ ​= 1 000 000 ​m2​ ​ 1 km (Not drawn to scale.) 1 km ​1​hectare (​ 1 ha)​ 100 m (Not drawn to scale.) 1​ ha = 10 000 ​m2​ ​ 100 m The dimensions of a rectangle are called length (​ ​l​)​and width (​ ​w)​ ​​. The area of a rectangle is given by the number of rows multiplied by the number of columns. Written as a formula, this looks like: ​ A=l×w w A = l × w​. This also works for numbers that are not integers. The area of a square is given by: ​A = l × l = ​l​2​ l A= l2 l BUILDING UNDERSTANDING 1 For this rectangle drawn on a ​1 cm​grid, find each of the following. a the number of single ​1 cm​ squares b the length and the width c length ​×​ width 2 For this square drawn on a centimetre grid, find the following. a the number of single ​1 cm​ squares b the length and the width c length ​×​ width 3 Count the number of squares to find the area in square units of these shapes. a b c Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  682 Chapter 10 Measurement 4 Which unit of area (​​mm​2​ ​, ​cm​2​ ​, ​m2​ ​, ha​or ​km​2​ ​) would you choose to measure these areas? Note that ​1 ​km​2​ ​is much larger than ​1 ha​. a area of an ​A4​piece of paper b area of a wall of a house c area of a small farm d area of a large desert e area of a large football oval f area of a nail head Example 10 Counting areas Count the number of squares to find the area of the shape drawn on this centimetre grid. SOLUTION EXPLANATION 6​ ​cm​2​ ​ There are ​5​full squares and half of ​2​squares in 1 the triangle, giving ​1​ more. of 2 = 1 2 Now you try Count the number of squares to find the area of the shape drawn on this centimetre grid. Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10E Units of area and area of rectangles 683 Example 11 Finding areas of rectangles and squares Find the area of this rectangle and square. a b 4 mm 2.5 cm 10 mm SOLUTION EXPLANATION a Area = l × w The area of a rectangle is the product of the ​ ​ =​​ 10 × ​4​ ​ length and width. ​ = 40 ​mm​2​ ​ b Area = ​l​2​ The width is the same as the length, so ​ ​ =​ 2.​​52​ ​ ​ A = l × l = ​l​2​ ​   ​ ​ ​= 6.25 ​cm​2​ ​ ​(2.5)​2​ ​= 2.5 × 2.5 Now you try Find the area of this rectangle and square. a b 3 cm 1.5 km 8 cm Exercise 10E FLUENCY 1, 2, 3​​(1​/2​)​,​ 4–6 2, 3​(​ 1​/2​)​, 4, 5, 7 2–3​(​ 1​/2​)​, 4, 5, 7, 8 Example 10 1 Count the number of squares to find the area of these shapes on centimetre grids. a b Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  684 Chapter 10 Measurement Example 10 2 Count the number of squares to find the area of these shapes on centimetre grids. a b c d Example 11 3 Find the area of these rectangles and squares. Diagrams are not drawn to scale. a b c 2 cm 10 cm 11 mm 20 cm 3.5 cm 2 mm d e f 5m 1.2 mm 2.5 mm g h i 0.8 m 17.6 km 0.9 cm 1.7 m 10.2 km 4 Find the side length of a square with each of these areas. Use trial and error if you are unsure. a ​4 ​cm​2​ ​ b ​25 ​m2​ ​ c ​144 ​km​2​ ​ Essential Mathematics for the Australian Curriculum ISBN 978-1-009-37279-4 © Greenwood et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  10E Units of area and area of rectangles 685 5 There are ​10 000 ​m2​ ​in one hectare (​ ha)​. Convert these measurements to hectares. a ​20 000 ​m2​ ​ b ​100 000 ​m2​ ​ c ​5000 ​m2​ ​ 6 A rectangular soccer field is to be laid with new grass. The field is 1​ 00 m​long and ​50 m​ wide. Find the area of grass to be laid. 7 Glass is to be cut for a square window of side length ​50 cm​. Find the area of glass required for the window. 8 Two hundred square tiles, each measuring 1​ 0 cm​ by ​10 cm​, are used to tile an open floor area. Find the area of flooring that is tiled. PROBLEM-SOLVING 9, 10 9–11 9, 11, 12 9 a A square has a perimeter of 2​ 0 cm​. Find its area. b A square has an area of ​9 ​cm​2​ ​. Find its perimeter. c A square’s area and perimeter are the same number. How many units is the side length? 10 The carpet chosen for a room costs ​$70​ per square metre. The room is rectangular and is ​6 m​ long by ​5 m​wide. What is the cost of carpeting the room? 11 Troy wishes to paint a garden wall that is ​11 m​ long and 3​ m​high. Two coats of paint are needed. The paint suitable to do the job can be purchased only in whole numbers of litres and covers an area of ​15 ​m2​ ​per litre. How many litres of paint will Troy need to purchase? 12 A rectangular area of land measures ​200 m​by ​400 m​. Find its area in hectares. REASONING