Measuring - Past Paper PDF

 % Measuring Inline skating is a branch of the skate sport that originated from roller skating. Especially in South America, this sport is extremely popular. In the Netherlands,...

 % Measuring Inline skating is a branch of the skate sport that originated from roller skating. Especially in South America, this sport is extremely popular. In the Netherlands, too, there is a lot of inline skating, for example as a summer alternative to ice skating. The only indoor competition skating rink in the Netherlands is in Heerenveen. The track is 200 metres long and consists of two straight sections and two semi-circles. The radius of the inner track is 13 metres, and the width of the track is 7 metres. With these data you can calculate the area of the track, and the length of the straight sections. © Noordhoff Uitgevers bv  Prior Knowledge Units Theory A Units There are many things you can measure. For example, length, time, temperature and speed. Temperature can be measured in degrees Celsius. You could measure that the temperature in the classroom is 22 °C, for example. A degree Celsius is a unit of temperature. Length can be measured in centimetres or metres. For example, you could measure that Laila is 161 cm tall, and Paul 1.58 metres tall. Centimetres and metres are units of length. 1 See the theory above. Give a few more examples of units of length. 2 Write down three units related to measuring a time c area b weight d volume 3 a In the following two sentences the units are missing. Which units are involved? ‘He doesn’t care about a few pounds more or less.’ ‘He exceeded 80 through that bend.’ b On the signs I, II, III and IV below the units are missing. Write down the correct unit for each sign. figure 7.1 4 In the situations below, note down the unit that fits best. I There is a rain barrel in Mr Goose’s garden. He wonders how much water is in it. Il Mr. Goose wants to put a fence around his garden. He wants to know how much fencing he needs. Ill He will also want to get new rolls of turf in his garden. He wants to know how much he needs. 58 Chapter 7 Measuring © Noordhoff Uitgevers bv  7.1 Length Learning objectives You can convert units of length. You can calculate to scale. s In this chapter you are allowed to use the calculator. 01 See figure 7.2. 3®* a What unit is used here? b How many metres is it to Utrecht? figure 7.2 Theory A Units of length Above you can see the historical signposting of the ANWB. The number behind the place name indicates the distance. This distance is measurable. Something that you measure is called a quantity. In this case the quantity of the distance has been measured in the unit of length called kilometre. Other frequently used units of length are millimetre (mm), centimetre (cm) and metre (m). Below the different units of length are shown from biggest to smallest. Each jump to the right means x 10, so 1 km = 10 hm. Each jump to the left means 10, so 5 dm = 0.5 m. UNITS OF LENGTH The international standard unit of distance is metre. The units of length above are derived from there. The k in km stands for ‘kilo’ and means 1000. So 1 km = 1 kilometre = 1000 metres. k = kilo = 1000 da = deca = 10 c = centi = 0.01 h = hecto = 100 d = deci = 0.1 m = milli = 0.001 © Noordhoff Uitgevers bv 7.1 Length 59  km hm dam m dm cm mm Each jump to the right means x 10. 1 hm = 10 dam 5.2 km = 5200 m 58 cm = 580 mm 8.43 dam = 843 dm Each jump to the left means 10. 1 dm = 0.1 m 78.3 dam = 0.783 km 826 hm = 82.6 km 25 mm = 0.025 m Learning objective You can convert units of length. 2 Complete the sentence by entering the correct unit of length, CJ a The Afsluitdijk is more than 30... long. b A jeweller sells necklaces with lengths from 35 to 80... c A football field is about 70... wide. d A jeweller sells rings with diameters from 12 to 22... 3 Fill in. CJ a 78 dm =... mm d 720 m =... km g 8000 cm =... m b 7080 mm =... m e 8.7 dam =... km h 72 m =... mm c 8.21 hm =... dm f 160 dm =... hm i 0.3 m =... hm 4 2.7 hm + 12 m + 3.9 dam = 270 m + 12 m + 39 m = 321 m ®* Calculate in the same way. a 4 km + 18 hm + 27 dam =... m b 5.2 km + 3 dam + 16 m =... hm c 4.73 m + 18 cm + 0.3 dm =... m d 8.2 m + 32 dm + 12 cm =... m e 6.3 hm + 21 dam + 700 dm =... km f 820 mm + 7 cm + 3 dm =... m A5 Karen is taking swimming lessons. She trains in a 25-metre pool. tJ®* Monday, she swims 36 laps, Wednesday she swims 325 metres more, and Friday she swims six more laps than Wednesday. How many kilometres does she swim in total? Dit bestand is uitsluitend bedoeld voor het gebruik met dyslexiesoftware en mag voor geen andere doeleinden worden gebruikt. Dit boek heeft een twee sterrenkwaliteit, d.w.z. dat de leesvolgorde door het softwareprogramma is bepaald, waardoor niet altijd de juiste leesvolgorde kan worden gegarandeerd 60 Chapter 7 Measuring © Noordhoff Uitgevers bv en niet alle correct wordt voorgelezen.  A6 Read the article. ®* The Hanzelijn is a railway between Lelystad and Zwolle. The double-track section was put into operation in 2012. The construction cost 1.05 billion euros. Along the 45 km long stretch 1600 masts have been erected. 13 000 rails were used. The rails are supported by a sleeper every 80 cm. The line is suitable for speeds up to 200 km per hour. On average, 17000 travellers a day board the train at one of the four Hanzelijn stations. In addition, 4 million tonnes of freight are transported on the Hanzelijn every year. (1 ton = 1000 kg) a The masts are on both sides of the railway. Calculate the average distance between two masts on the same side of the track. b Calculate the length of a steel rail in cm. Round to the nearest integer. c How many railway sleepers does the route have? d How many people per year board on the Hanzelijn stations? Round to hundreds of thousands. e Between Flevoland and Overijssel, a train travels at a speed of 100 km per hour through the Drontermeertunnel. This takes 47 seconds. What is the length of the tunnel? Round to tens of metres. □ Check I can convert units of length. Not quite mastered this learning objective yet? Then study theory A and do exercise LI. LI J Fill in. a 12dm =... mm c 15.2 dam =... dm e 1800 cm =... hm b 1600 cm =... m d 3.1 hm =... m f 80 m =... km 07 During his holiday in Dubai, Adam bought a keychain □ ® * with a model of the Burj Khalifa. The height of the model is 8.3 cm. The actual height of the Burj Khalifa is 830 m. Fill in. a The height of the model is... times as small as in reality. b 1 cm of the model corresponds to... cm in reality. The Burj Khalifa. © Noordhoff Uitgevers bv 7.1 Length 61  Theory B Scale The Burj Khalifa keychain in exercise 7 is a scale model. The scale is 1:10 000. This means that 1 cm of the model represents 10000 cm. All the real dimensions are 10000 times bigger than the dimensions of the model. You could also say that all the model’s dimensions are 10000 times smaller than the real dimensions. For scale 1:25, all dimensions of the model are 25 times smaller than in reality. Scale 1:200 means that all the dimensions are 200 times bigger in reality than in the model. Learning objective You can calculate to scale. 8 Rail manager ProRail has recreated the Hanzelijn in Ö® miniature in collaboration with Miniworld Rotterdam on a scale of 1:160. The railway bridge over the IJssel is part of the Hanzelijn. On the right you see the scale model of the railway bridge. a The scale model is 585 cm long. How many metres is the bridge in reality? b How many cm is the scale model of a train with an figure 7.3 actual length of 140 metres? A9 A scale model of the Fokker 50 plane is 63 cm long. Q® * In reality, the length of the aircraft is 25.2 metres. a Calculate the scale of the model. b The wingspan is the distance between the two wing tips. In reality, the wingspan of the aircraft is 29 metres. How many cm is the wingspan of the scale model? □ Check I can calculate to scale. Not quite mastered this learning objective yet? Then study theory B and do exercise L2. z L2j A model of the Eiffel Tower is made to scale 1:1600. a The model is 20 cm tall. What is the actual height of the Eiffel Tower in metres? b The second floor of the Eiffel Tower is at a height of 115 metres. How high is the second floor in the model in cm? Round to the nearest tenth. 62 Chapter 7 Measuring © Noordhoff Uitgevers bv  7.2 Area Learning objectives You can determine the area of a figure on grid paper. You can convert area units. You can calculate the area of a figure that can be divided into rectangles. 010 Mr. Lambert is going to lay square carpet tiles of □ ®* 1 by 1 metre on his living room floor. The living room is 8 by 5 metres. a How many carpet tiles does he need? b Skirting boards will be installed along the edges of the floor. How many metres of skirting board does he need? figure 7.4 Theory A Perimeter and area The perimeter of a figure is a path that outlines a shape or its length. For example, the perimeter of the heptagon on the side is 35 + 15 + 15 + 16 + 22 + 9+16= 128 mm. The perimeter of a figure is the sum of the length of its sides. figure 7.5 The dimensions are in mm. Area has to do with coverage. The area of a figure indicates how many times the unit of area fits in the figure. A frequently used unit of area is the square centimetre. Notation: cm2. 1 cm2 is the area of a square which is 1 by 1 cm. 5 whole and 2 half-squares of 1 cm2 fit on the figure shown on the right. This means its area is 6 cm2. The area of a figure indicates how many times the unit of area fits in it. Learning objective You can determine the area of a figure on graph paper. © Noordhoff Uitgevers bv 7.2 Area 63  12 Estimate how many cm2 the area of the figure on O®* the right is. figure 7.9 A13 A rectangular piece of land measuring 120 □ ® * by 80 metres is divided into seven pieces. See the figure on the right. a How many m2 is the area of a grid square in the figure on the right in reality? b Which piece has the largest area? Calculate that area. c Calculate the area of the smallest piece. figure 7.10 E14 The vertices K, L, M and N are the centres of the sides ® * of square ABCD. The vertices P, Q, R and S are the centres of the sides of square KLMN. See the figure on the right. Square PQRS has an area of 6 cm2. How much larger is the area of ABCD than the area of KLMN in cm2? figure 7.11 64 Chapter 7 Measuring © Noordhoff Uitgevers bv  E15 In the figure on the right you see Mrs Glover’s ® * vegetable garden. All the corners in the figure are right angles. Calculate the perimeter of the vegetable garden. figure 7.12 The dimensions are in metres. □ Check I can determine the area of a figure on grid paper. Not quite mastered this learning objective yet? Then study theory A and do exercise L3. / L3J How many cm2 is the area of the figure on the right? figure 7.13 016 On the right you see a piece of graph paper. A red square 21 ® * of 1 by 1 cm is drawn. a How many squares of 1 by 1 mm fit inside the red square? b Fill in. 1 cm2 =... mm2 figure 7.14 Theory B Units of area Three rows of four squares of 1 cm2 each fit in a 4 by 3 cm rectangle. This means the area is 4 3 = 12 cm2. This is an example of the following rule. area of a rectangle = length widthj In exercise 16 you saw that 1 cm2 = 100 mm2. It follows that the area of the rectangle in figure 7.15 figure 7.15 equals 1200 mm2. To convert the unit of length in cm2 to the unit of length in mm2, you have to multiply by 100. CLAfy., © Noordhoff Uitgevers bv 7.2 Area 65  Below are some units of area from biggest to smallest. UNITS OF AREA Each jump to the right means x 100. 1 km2 = 100 hm2 2.8 m2 = 280 dm2 0.7 dm2 = 7000 mm2 Twojumps to the right, so x 10000. Each jump to the left means + 100. 80 dm2 = 0.8 m2 ha = hectare = 10O are 32.5 are = 0.325 ha 3800 m2 = 0.38 hm2 Twojumps to the left, so -f 10000. Learning objective You can convert the units of area. 17 Enter the correct unit of length or area. Cl a The area of the Netherlands is 41 864... b The surface of a tray is 1480... c The Westerscheldetunnel is more than 6... long. d An allotment has an area of 75... e The Binnenhof in The Hague is approximately 1.75... f The area of a postage stamp is approximately 475... 18 Fill in. □ a 0.4 m2 =... cm2 d 33 km2 =... are g 0.35 m2 =... cm2 b 34 mm2 =... dm2 e 46are =...ha h 2250m2 = „. ha c 0.02 km2 =... m2 f 0.018 ha =... m2 i 83 dm2 =... are 66 Chapter? Measuring © Noordhoff Uitgevers bv  19 160 m2 + 1.2 ha = 160 m2 + 12000 m2 = 12160 m2 ®* Calculate in the same way a 50 cm2 + 2.5 dm2 =... cm2 b 2.4 are + 035 ha+ 150 m2 =... m2 c 80 mm2 + 0.3 dm2 =... cm2 d 2.81 m2 - 260 dm2 =... cm2 e 5 ha - 480 are + 120 dm2 =... m2 f 74 000 cm2 + 300 dm2 =... m2 20 a Calculate the area and perimeter of a rectangle measuring Add the correct unit. □ 7 by 4 cm. b Calculate the area and perimeter of a square with sides of 4 cm. c Calculate the area of a square with a perimeter of 48 cm. d Calculate the perimeter of a square with an area of 100 cm2. E21 In the square on the right with sides of 18 cm, seven ®* equal rectangles have been drawn. How many cm2 is the area of such a rectangle? E22 A large square consists of four equal rectangles and a * smaller square. The diagonals of the rectangles are 5 cm long, and the area of the large square is 49 cm2. See the figure on the right. How many cm2 is the area of the small square? © Noordhoff Uitgevers bv 7.2 Area  23 a A 4.8 km long cycle path is 1.5 metres wide. The construction costs □ are 48 euros per m2. Calculate the total construction cost. b Tiles measuring 60 by 60 cm will be installed in a courtyard measuring 12 by 4.80 metres. The tiles cost € 8.75 each. Calculate the total cost of the tiles. 24 Ivar made a rectangular table with a width of 80 cm, a length of 150 cm O® and a height of 75 cm. There will be a trim around the table top. The DIY store only sells the trim in 2-metre pieces. a How many of those pieces does Ivar need? b Ivar has just enough trim left for a square table. Calculate in dm2 the area of the table top of this table. A25 a A rectangular piece of land is 285 metres long Why do you have an Ö®* and has an area of 3.42 ha. The land is divided tment with oust grass? into allotments that are each 15 by 30 metres. How many allotments fit on the piece of land? Isn’t that beautiful, so nice and neat? The robotic b Wheat is sown on a field of 450 by 140 metres. lawnmower does all the work. 160 kg of wheat seed is needed per ha. The wheat seed costs 53 euros per bag of 20 kg. Calculate the total cost of the wheat seed. c In a basement, the four walls and the rectangular floor are painted. The floor is 62 by 74 dm, and the walls are 25 dm high. 8 m2 can be painted with a can of paint. How many cans of paint are needed? □ Check I can convert area units. Not quite mastered this learning objective yet? Then study theory B and do exercise L4. Fill in. a 4 dm2 =... cm2 c 1.5 ha =... m2 e 1800 are =... km2 b 16dm2 =... m2 d 3000 dm2 =... are f 7.2 mm2 =... dm2 026 In the figure on the right, the dimensions are in 6 □ ® * cm. To calculate the area, you can divide the figure into two rectangles. a Divide the figure into two rectangles. What are the dimensions of these rectangles? ----- b Calculate the area of each rectangle, then calculate the area of the whole figure. 15 figure 7.18 68 Chapter 7 Measuring © Noordhoff Uitgevers bv  Theory C Calculating areas Sometimes you can divide a figure into rectangles. In that case you derive the area of the entire figure by adding up the area of each of the rectangles. See the example below. Learning objective You can calculate the area of a figure you can divide into rectangles. Example In the figure on the right the dimensions are in cm. Calculate the area of the figure. Approach Divide the figure into rectangles and label the dimensions involved. figure 7.19 Solution See the figure on the right, area = area I + area II =4 1+5 2 = 4+10=14 cm2 Add the unit. R27 You can also calculate the area of the figure in the □ ®* example above in another way. See the figure on the right. 2 a What is the area of rectangle ABCD3 b What do you need to subtract from this to get the A area of the yellow figure? figure 7.20 c Check if you get the same area of the yellow figure in this way. 28 Calculate the area of the figures below. Ü®* figure 7.21 The dimensions are in cm. © Noordhoff Uitgevers bv 7.2 Area 69  29 On the right you see the floorplan of the ground Ü® floor of a house. The living room is carpeted. The price for the carpet is 35 euros per m2. a What are the total costs for the carpet? Skirting boards will be installed along the edge of the living room. The skirting boards cost €2.40 per metre. b How much do the skirting boards cost? You do not have to worry about doors. figure 7.22 The dimensions are in metres. A30 A swimming pool is 26 by 12 metres. There will O ® * be a tile path around the pool that is 120 cm wide throughout. The tiles are 40 by 40 cm and cost €0.89 each. See figure on the right. Calculate the total cost of the tiles. figure 7.23 A31 On the right you see the floor plan ® * of an apartment on a scale of 1:250. a Measure the balcony and calculate the area of the balcony in reality. b The bedroom is 2.4 metres high. Tanya is going to paint the left wall of this room. With a can of paint, she can paint 2.6 m2. How many cans does she need? figure 7.24 Scale 1:250. □ Check I can calculate the area of a figure that can be divided into rectangles. Not quite mastered this learning objective yet? Then study theory C and do exercise L5. Calculate the area of the figure on the right. figure 7.25 The dimensions are in cm. 70 Chapter 7 Measuring © Noordhoff Uitgevers bv  7.3/The Circle Learning objectives You can calculate the circumference of a circle. You can calculate the area of a circle. You can calculate the area and circumference of a circle sector. 032 You can measure the circumference of a round object J ® * with a wire or a rope. See the picture on the right. a Measure the circumference of a round object of your choice in the same way. Also measure the diameter.. ,. circumference _ ,. b Calculate the quotient —j;---- -------- of your object n diameter of choice. c Compare your answer to question b with that of some of your classmates. What do you notice? figure 7.26 Theory A Circumference In exercise 32 you saw that the quotient of a round object’s circumference. , t ---- ------------- is about 3.14. If you were to measure very diameter carefully, you would arrive at 3.14159... This goes for every circle. This gives the formula circumference = 3.14159... diameter. For the number in the formula we use the Greek letter it. So n = 3.14159... n is pronounced as ‘pi’. n Your calculator has a 0 button. Have a look at how 3.1 Hl 5 9265 H many digits your calculator shows when you hit a:. By using a the formula for the circumference becomes circumference = a diameter. Sometimes you know the radius of a circle. You can calculate the diameter with the formula diameter = 2 radius. circumference = n diameter diameter = 2 radius Learning objective You can calculate the circumference of a circle. © Noordhoff Uitgevers bv 7.3 The Circle 71  Example The front wheel of Abdel’s bike has a radius of 33 cm. a Calculate the circumference of the wheel in cm. Round to the nearest integer. b Abdel cycles to school with a speed of 18 km per hour. It takes him 10 minutes. How many rotations did the wheel make? Solution With the EJ3 a diameter = 2 33 = 66 cm button you get the circumference = n 66 ~ 207 cm outcome that was b 18 km per hour, so 3 km in 10 minutes. calculated last. 3 km = 300000 cm 300000-207.34...- 1447 Use the unrounded answer of question a. So 1447 rotations. Enter O®O®®® O CO 6- R33 See the solution of question b in the example. O®* Why was the unrounded answer to question a used for the calculation? 34 a A circle has a diameter of 16 cm. Ö Calculate the circumference in cm. Round to the nearest tenth. b A circle has a radius of 38.6 cm. Calculate the circumference in cm. Round to the nearest hundredth. 35 The rear wheel of Nina’s bicycle has a diameter of 52 cm. CJ Nina cycles to her grandmother in 20 minutes. She cycles 15 km per hour. How many rotations has the wheel made? 36 The round pond in Pondview park has a diameter of 91 metres. □ ®* a How many metres is the length of the pond edge? Round to the nearest tenth. There is a fence three metres from the pond edge. b Calculate the length of the fence in metres. Round to the nearest tenth. For question c, use c Melvin can walk along the fence in five minutes. the unrounded answer Calculate his speed in kilometres per hour. Round to the to question b. nearest tenth. 37 A round table top has a diameter of 85 cm. a®* The table top will have a trim all around that costs € 1.95 per metre. How much do you have to pay for the entire trim? 72 Chapter 7 Measuring © Noordhoff Uitgevers bv  A38 The wheels of Christian’s mountain bike have a □ diameter of 29 inches. One inch corresponds to 25.4 mm. a Christian makes a trip on his mountain bike. When he returns, his bike computer shows that he has ridden 30.82 km. How many times did the wheels rotate on this trip? Round to the nearest tenth. b The manufacturer recommends changing the tires after one million rotations. How many kilometres is that? Round to the nearest ten. A 39 The Ferris wheel ‘High Roller’ was built in Las Vegas in 2014. The wheel is 176.6 metres high and has a diameter of 158.5 metres. There are 28 gondolas that can carry 40 people each. During the opening hours, the wheel rotates continuously at a speed of 30.5 cm per second. Calculate how many people can enter the High Roller per hour. E40 The diameter of the circle on the right is 10 cm. ® Exactly four rows of four equally sized rectangles fit into the circle. What is the perimeter of the grey area in cm? E41 The big hand on a church clock is 3 metres long and the small hand * 2 metres. At some point, the tip of the small hand travelled 100 metres. Calculate without a calculator how many metres the tip of the big hand has travelled in the same time. □ Check I can calculate the circumference of a circle. Not quite mastered this learning objective yet? Then study theory A and do exercise L6. ri L6j a Calculate in cm the circumference of a circle with a diameter of 25 cm. Round to the nearest tenth. b Calculate in cm the circumference of a circle with a radius of 25 cm. Round to the nearest tenth. c Calculate in metres the circumference of a circle with a radius of 143 dm. Round to the nearest integer. © Noordhoff Uitgevers bv 7.3 The Circle 73  042 On the right you see a dartboard without an □ ® * edge, with 20 sectors. You can cut the sectors apart and put them together as shown in the bottom figure. You can see that one sector has been cut in half, a In ABCD you can recognize a special quadrilateral. Which one is it? b Is AD slightly longer, slightly shorter, or a equal to the radius of the dartboard? c According to Felicia, AB is approximately half the circumference of the board. Do you agree with Felicia? Why? d According to Felicia, the area of rectangle ABCD is approximately equal to the area of the dartboard. figure 7.28 Do you agree with her? Why? Theory B Area of a circle In figure 7.28 the circle has been divided into pieces. These pieces were used for a figure that almost equals rectangle ABCD. From this rectangle ABCD, Remember that AD is the radius of the circle circumference of a circle = 7t diameter. AB is about half the circumference, so AB ~ it radius +........................ area ABCD =AB AD ~ it radius radius = it radius2. We usually indicate the Now you understand that the radius of a circle by the area of a circle = it radius2. letter r. You can use this formula to calculate the area of a circle r stands for radius. when you know the radius of the circle. Then you get: area of a circle = n radius2J area = nr2 diameter = 2r circumference = 2tu- Learning objective You can calculate the area of a circle. Example The diameter of a round table is 82.4 cm. Calculate the area of the table top in cm2. Round to the nearest integer. Solution First calculate the radius, because the radius = 82.4 2 = 41.2 cm P formula area of a circle = it radius2 area = % 41.22 » 5333 cm2 does not contain the diameter. 74 Chapter 7 Measuring © Noordhoff Uitgevers bv  43 a A circle has a radius of 8 cm. Ö Calculate the area in cm2. Round to the nearest tenth. b A circle has a radius of 27 cm. Calculate the area in cm2. Round to the nearest integer. c Calculate the area in m2 of a circle with diameter 173 dm. Pay attention to Round to the nearest integer. the units. 44 A round lake has a diameter of 520 metres. □ Calculate the area of its surface in ha. Round to the nearest integer. 45 Jolene’s hobby is making earrings. She makes earrings Ct ® * from square metal plates of 3 by 3 cm, as shown in the figure on the right. The diameter of the small circle is equal to the radius of the large circle. a Calculate the area of the earring in cm2. Round to the nearest hundredth. b Jolene makes twelve of these earrings in a week. How many cm2 of metal does she have left as waste this week? Round to the nearest integer. figure 7.29 CTD E46 In the figure on the right, you see three circles, each ® * with a diameter of 5 cm. Calculate what percentage of the figure is coloured blue. figure 7.30 47 A roundabout consists of a roadway with a width □ of 4.5 metres and a circular inner area with a diameter of 16 metres. a There will be a fence around the inner area. What is the length of the fence in metres? Round to the nearest hundredth. b The inner area is sown with grass. 25 grams of grass seed is needed per m2. Calculate how many kg of grass seed are needed for the inner area. Round to the nearest tenth. A48 See the roundabout on the right. figure 7.31 Cl ® * a Calculate the area of the roadway on the roundabout in m2. Round to the nearest tenth. b While road works are being carried out, a white line is applied in the midlSeTrf the roadway of the roundabout.The striping machine travels at a speed of 25 cm per second. How many minutes does it take to apply the line? Round to the nearest integer. © Noordhoff Uitgevers bv 7.3 The Circle 75  A49 The inline skate track in the figure on the □ ® * right consists of straight sections and semicircles. The radius of the inside bend is 13 metres, and the track is 7 metres wide The length of the track is 200 metres. This is measured at 50 cm from the inner edge of the track. See the red line in the figure, figure 7.32 a Calculate the length of a straight section in metres. Round to the nearest tenth, b Calculate the area of the track in m2. Round to the nearest integer. E50 The circumference of the centre circle of a football field is 57.5 metres. ® * Calculate the area of the centre circle in m2. Round to the nearest integer. □ Check I can calculate the area of a circle. Not quite mastered this learning objective yet? Then study theory B and do exercise L7. / L7j a Calculate in cm2 the area of a circle with a radius of 40 cm. Round to the nearest integer. b Calculate in cm2 the area of a circle with a diameter of 40 cm. Round to the nearest tenth. c Calculate in m2 the area of a circle with a radius of 127 dm. Round to the nearest integer. For thousands of years people have been fascinated by the number it. The oldest 22 known approximation of n, comes from Egypt and dates back to about 2600 B.C. Around 250 B.C. the Greek mathematician Archimedes demonstrated that the 223 22 actual value of 7t lies between nyp (~ 3.14085) and -y(~ 3.14286). The Dutchman Ludolph van Ceulen caused a sensation in 1596, by calculating no less than 35 decimals of it correctly, which was an enormous achievement for that time. He was so proud of his achievement that he had the decimals engraved on his tombstone. In 1874, before the advent of calculators, the record stood at 527 decimals. In 1882, the German Von Lindemann proved that n has an infinite number of decimals with no regularity. Nowadays you can calculate thousands of decimals of % with an app like Calculate Pi. Memory artist Rick de Jong from Oss, however, doesn’t need an app. On 1 April 2015 he became the European record holder by flawlessly naming the first 22 612 decimals of n. His record was broken in 2020 by the Swede Jonas von Essen, who only made a mistake after 24 063 decimals. 76 Chapter 7 Measuring © Noordhoff Uitgevers bv  051 A round pie is cut into pieces. See the figure on the n®* right. a Alex gets the piece labelled ‘30°’. What part of the pie does he get? b Ann gets the piece labelled ‘45°’. What part of the pie does she get? figure 7.33 Theory C Sectors An arc is part of a circle. On the figure to the right, arc AB is coloured red. Inside the figure, the circle sector is coloured white. A circle sector is an area enclosed by two rays and an arc. The angle between the rays is called the central angle. The central angle of the entire circle is 360°. The area of a sector with central angle 60° is | of the area of the circle. Likewise, the area of the sector with central angle 43 43° is 777 of the area of the circle. 360 a° is 777 of the area of the circle. 360 So the area of the sector with central angle a° is 777 area = 777 it radius2. 360 360 The length of an arc of a sector with central angle a° is — — of the circumference. So the length of the arc is 777 circumference circle = 777 it diameter. 360 360 circle sector with central angle a° area of a sector = 777 it radius2 360 length of an arc = it diameter Learning objective You can calculate the area and circumference of a circle sector. © Noordhoff Uitgevers bv 7.3 The Circle 77  Example The radius of the circle on the right is 12 cm, and the central angle of the green sector is 140°. a Calculate the area of the green sector in cm2. Round to the nearest integer. b Calculate the perimeter of the white sector in cm. Round to one decimal place. Solution figure 7.35 140 , a area = —77 n 122 = 176 erm 360 140 b perimeter = 2 12 + —777 n 24 ~ 53.3 cm perimeter of a sector = 360 2 radius + length of arc diameter R52 See example. □®* a Calculate the area of the green sector in cm2. Round to the nearest integer. b Calculate the perimeter of the green sector in cm. Round to the nearest tenth. 53 A sector of a circle with diameter 28 cm has a central angle of 78°. Q® a Calculate the area of the sector in cm2. Round to the nearest integer. b Calculate the perimeter of the sector in cm. Round to the nearest tenth. 54 In the figure on the right, the dimensions are in cm. Q® a Calculate the area of the green area in cm2. Round to the nearest tenth. b Calculate the perimeter of the green area in cm. Round to the nearest tenth. figure 7.36 55 On the right you can see which area of a snowy ® * window of a car is being wiped clean by the window wiper. Calculate the size of that area in cm2. Round to the nearest integer. figure 7.37 78 Chapter 7 Measuring © Noordhoff Uitgevers bv  A56 In shot put, an athlete throws a ball while □ ® * standing in a circle with a diameter of 4.2 metres. The ball must land within a sector with a central angle of 35°. If the ball goes outside the sector, the throw is invalid. The distance of a valid throw is measured from the edge of the circle. See the figure on the right, which assumes a maximum throw of 26 metres. a Arc AB will be painted white. figure 7.38 How many metres will be painted? Round to the nearest tenth. b In the shot put sector outside the circle, a new layer of gravel is applied. How many m2 of gravel is needed? Round to the nearest hundredth. A57 Calculate in cm2 ® * the area of the coloured figures on the right. Round to the nearest tenth. figure 7.39 E58 See the party hat on the right. The net of the party hat is a * circle sector. Calculate the area of this net in cm2. Round to the nearest integer. 12 cm figure 7.40 □ Check I can calculate the area and perimeter of a circle sector. Not quite mastered this learning objective yet? Then study theory C and do exercise L8. / L8J A sector of a circle with diameter 78 cm has a central angle of 127°. a Calculate the area of the sector in cm2. Round to the nearest integer. b Calculate the perimeter of the sector in cm. Round to the nearest tenth. © Noordhoff Uitgevers bv 7.3 The Circle 79  7 A Volume Learning objectives You can convert units of volume. You can calculate the volume and surface area of a cuboid. 059 On the right you see a box that has the Ö®* shape of a cuboid. The dimensions are 4 by 3 by 2 cm. The bottom is filled with cubes with edges of I cm. a How many cubes are there on the bottom? b How many cubes do you need to fill the box? figure 7.41 The dimensions are in cm. Theory A Units of volume Solids like a cuboid and cube have a volume. Volume has to do with filling. The volume of a solid indicates how many times the unit of volume fits in it. A frequently used unit of volume is the cubic centimetre. Notation: cm3. cm3 is called a cubic 1 cm3 is the volume of a cube of 1 by 1 by 1 cm. centimetre. Inside the cuboid of exercise 59, you can fit 24 cubes of 1 cm3. So the volume is 24 cm3. The volume of a solid indicates how many times the unit of volume fits in it. You can calculate the volume of a cuboid with the following formula volume of a cuboid = length width height. For the cuboid in exercise 59 this gives volume = 4 3 2 = 24 cm3. volume of a cuboid = length width height Learning objective You can calculate the volume of a cuboid. 80 Chapter 7 Measuring © Noordhoff Uitgevers bv  Frequently used units of volume are m3, dm3 and cm3. 1 dm3 = 1 litre The litre (1) is also often used. 11=1 dm3 You can divide a litre into decilitres, centilitres and millilitres. 1 1 = 10 dl = 100 cl = 1000 ml Below are different units of volume from biggest to smallest. UNITS OF VOLUME X 1 000000 x 10 x 10 x 10 Learning objective You can convert units of volume. Examples 3.2 hm3 = 3200 dam3 8.5 dl = 85 cl 5 dm3 = 5 1 800 hm3 = 0.8 km3 72 ml = 0.72 dl 7 cl = 0.07 1 = 0.07 dm3 0.08 m3 = 80000 cm3 0.023 I = 23 ml 70 cl = 700 ml = 700 cm3 60 Fill in. a 8 dm3 =... cm3 f 5 1 =... dm3 b 2 1 =...cl g 1700 cm3 =... 1 c 60000 mm3 =... dm3 h 2.55 dm3 =... dl d 75ml =... di i 1.5 dl =... mm3 e 0.51 hm3 =... m3 j 0.7 I =... cm3 61 12 dm3 + 35 dl = 12 1 + 3.5 1= 15.5 1 ® * Calculate in the same way. a 300 cm3 + 2 dm3 =... cm3 b 0.2 dam3 + 130 m3 =... dm3 c 2 dl + 12 ml =... cl d 0.5 dm3 + 32 cm3 + 6 dl =... cl © Noordhoff Uitgevers bv 7.4 Volume 81  □ Check I can convert units of volume. Not quite mastered this learning objective yet? Then study theory A and do exercise L9. Fill in. a 3 dm3 =... cm3 e 1.61 =... cl b 4000 dm3 =... m3 f 15 ml =... dl c 12 000 dam3 =... km3 g 1.2dam3 =... 1 d 3 cm3 =... 1 h 5500 cl =... m3 62 A cuboid is 1.5 dm long, 4 cm wide and 5 cm high. Ö How many ml is the volume of that cuboid? First convert the dimensions 63 A swimming pool is 9 m long, 4 m wide and 150 cm to the same unit. Ü®* deep. a Calculate the volume of the pool in m3. b The pool is being filled. Per minute, 25 litres of water run from the tap. How many hours does it take to fill the pool? A64 24000 m3 of sand is needed to prepare a new residential area for □ ®* construction. The sand is brought in by trucks. The body of a truck is cuboid-shaped and has the dimensions 70 by 24 by 14 dm. a How many times does the truck need to drive to deliver the sand? b 42 000 kg of sand is loaded per truck. How many kilos does a dm3 of sand weigh? Round to the nearest hundredth. c Twelve wheelbarrows are needed for 1 m3 of sand. Would it take more or less than 100000 wheelbarrows to deliver 24000 m3 of sand? E65 During a downpour, 15 litres of water per square metre came down. ® * How many cm has the water level in a rain barrel risen during that downpour? 66 See the cuboid on the right with a length of 5 cm, □ ®* width 2 cm and height 3 cm. In this exercise you need to calculate the surface area of the cuboid. You are going to use: The surface area of a solid is the area of its net. figure 7.42 The dimensions are in cm. a Draw the net of the cuboid. b Calculate the surface area of the cuboid by calculating the area of the net. 82 Chapter 7 Measuring © Noordhoff Uitgevers bv  67 A box without a lid has the following dimensions: ® length 35 cm, width 28 cm and height 2 dm. a Calculate the surface area of the box in cm2. b Calculate the volume of the box in litres. A68 The outside of the box on the right is getting a 4 ® * paint job. First there are two layers of primer and then a layer of varnish. The top and bottom are also painted. The paint is in cans that will cover 8 m2. a How many cans of primer and how many cans of varnish are needed? b The walls of the box are 15 mm thick. Calculate the volume of the box in litres. Round to the nearest tenth. figure 7.43 E69 Susan turns a cube into eight smaller cuboids by cutting ® * the cube three times. Then she divides the total surface area of the eight cuboids by the total surface area of the cube. What will Susan get as a result? figure 7.44 E70 See the cylinder on the right. ® * You calculate the volume of a cylinder with the formula cylinder volume = area of the base height. The height of a cylinder is 20 cm and the diameter of the base is 6 cm. a Calculate the volume in centilitres. Round to the nearest tenth. b Calculate the area in cm2. Round to the nearest integer. □ Check I can calculate the volume and the surface area of a cuboid. Not quite mastered this learning objective yet? Then study theory A and exercise 66 and do exercise LIO. z Lio] A cuboid is 20 dm long, 15 dm wide and 40 cm high. a Calculate the volume of the cuboid in litres. b Calculate the surface area of the cuboid in dm2. © Noordhoff Uitgevers bv 7.4 Volume 83  7.5 Views Learning objective You can make use of views. 071 Below you see a doghouse on the left. To the right, you can see the Ü®* doghouse from the front, from above and from the side on a scale of 1:50. front view figure 7.46 a You can measure that the height of the front view is 2.0 cm. How high is the doghouse in reality? b The doghouse is shortened by 20 cm at the bottom. Which view does not change? Theory A Views In exercise 71 you have seen three views of a doghouse: the front view, the top view and the side view. In the front view and the side view you can measure the height of the doghouse. The measured height is 2.0 cm. The scale is 1:50, so the real height is 50 2.0 = 100 cm, or 1 metre. The top view gives no information about the height of the doghouse. However, you can measure the length and the width from the top view of the doghouse. The measured length is 3.0 cm, so the real length is 50 3.0 = 150 cm, or 1.5 metres. The measured width is 2.5 cm, so the real width is 50 2.5 = 125 cm, or 1.25 metres. Learning objective You can make use of views. 84 Chapter 7 Measuring © Noordhoff Uitgevers bv  72 Below you see a sandpit with the three views next to it. Ö®* front view side view top view figure 7.47 Scale 1:50. a Calculate the area in dm2 of the back of the sandbox. The sandbox will get a lid that fits exactly on the sandbox. b To find out the width of the lid, you need to measure the side view. Why? c Calculate the area of the lid in m2. Round to the nearest tenth. 73 Below you see a house with a garage. You see different views. □ ®* front view top view figure 7.48 Scale 1:300. a Calculate the area of the front door of the house. b New roofing felt is applied to the roof of the house and garage. That costs 65 euros per m2 including labour. Calculate the total cost of applying the roofing felt. c Calculate in m3 the volume of the house and the garage together. You do not have to take the thickness of the walls into account. d The garage door has been stained. How many m2 is it? e The exterior walls of the garage are being painted. One litre of paint is enough for 6 m2. How many litres of paint are needed? © Noordhoff Uitgevers bv 7.5 Views 85  M4 Below you see a garden with an earthen wall and three views. 0®* front view side view top view figure 7.49 Scale 1:150. Grass is sown on the earthen wall. 35 grams of grass seed are needed per m2. One kg of grass seed costs 18 euros. Calculate the total cost for the grass seed. 75 Draw the front view, right side view and top 20 □ view of the object on the right on a scale of 1:20. figure 7.50 The dimensions are in cm. A76 Draw the front view, side view and top view Q®* of the house on the right on a scale of 1:200. You do not need to draw the windows and doors. The dormer is at 1 metre below the highest point of the roof. figure 7.51 The dimensions are in m. A77 On the right you see the top view of a ® * construction made of cubes with edges of 1 cm. Each box shows the number of cubes stacked on top of each other. side a Draw the front and the side view. b How many cubes can you remove at most so that none of the three views changes? c How many cubes can you add at most front so that none of the three views changes? figure 7.52 Three views belong to this construction. 86 Chapter 7 Measuring © Noordhoff Uitgevers bv E78 Below you see four cubes in which solids are drawn. For each solid in ® * figure a, find the corresponding front, side and top view from figure b. a A B C D E F G b figure 7.53 E79 On the right the front view of a cylinder is drawn on a scale * of 1:200. a Calculate the volume of the cylinder in m3. Round to the nearest integer. b Calculate the surface area of the cylinder in m2. Round to the nearest integer. figure 7.54 □ Check I can make use of views. Not quite mastered this learning objective yet? Then study theory A and do exercise LU. L11J Below you see a shed and next to it three views on a scale of 1:150. a Measure the height of the door in the front view. What is the actual height of the door? b Calculate the area of the shed’s window. c In which views can you measure the length of ABI d Calculate the area of the roof. © Noordhoff Uitgevers bv 7.5 Views 87  Mixed Exercises 1 Fill in. D a 7000 cm =... m d 21ha =...m2 g 40001 =...hl b 17 m =... mm e 500000 cm2 =... m2 h 2.2 1 =... dm3 c 3.7 m2 =... dm2 f 5.2 I =... ml i 80 cl =... cm3 2 Fill in. ®* a 72 dam + 0.3 km + 1700 dm =... m b 4.8 m2 + 6052 cm2 + 38000 mm2 =... dm2 c 270 ha + 850 are — 750000 m2 =... km2 d 3600 ml + 0.85 dm3 + 26 dl =... 1 e 2.35 dm3 + 340 ml + 745 cl =... cm3 3 On the right you see the map of a plot. □ a Calculate the area of the plot. b The plot will be surrounded by a fence with a height of 50 cm. Calculate how many m2 of fence are needed. 70 figure 7.56 The dimensions are in metres. 4 The parents of Irene and Jane have a round ®* swimming pool in the garden. The diameter of the pool is eleven metres. Irene swims ten laps along the edge of the pool. She always stays one metre from the edge. She takes six minutes to complete these ten laps. Jane walks with Irene and always stays fifty cm from the edge of the pool. Jane walks with an average speed that is higher than the average speed at which Irene swims. What is the difference in speed in kilometres per hour? Round to the nearest hundredth. 5 A total of 300 000 m3 of sand is needed for the construction of a section Ü® * of motorway in Zeeland. One third of the sand is delivered in three weeks by ten trucks over a 10 km long route. The body of a truck is 6.4 m by 25 dm by 125 cm. Calculate how many times each truck must drive back and forth. 88 Chapter 7 Measuring © Noordhoff Uitgevers bv  6 On the right you see a door surrounded by a 3®* concrete frame that is 20 cm wide. The upper part of the door has the shape of a semicircle. a A weatherstrip is applied all around the door, including the bottom. Calculate how many cm of weatherstripping is needed. Round to the nearest integer. b The concrete frame is painted. How many m2 is it? Round to the nearest tenth. figure 7.57 7 A swimming pool of 24 by 15 metres is 2.5 metres deep. A tiled path that U® * is 120 cm wide will be built around the pool. The tiles are 30 by 30 cm and cost 75 cents each. a Calculate the total cost of the tiles. If necessary, make a sketch of the situation. b The bottom and sides of the pool are scrubbed. How many m2 is being scrubbed? c The pool is filled up to 15 cm below the rim. During filling, 150 litres of water enter the pool per minute. How many days does it take to fill the pool? Round to the nearest integer. 8 See the house and garage on the right. □ ®* a Draw the front, right side and top views on a scale of 1:200. You do not need to draw windows and doors. b Calculate in m2 the total area of the two sloping roof parts and the roof of the garage. figure 7.58 The dimensions are in metres. 9 On the right you see a design for a circular Ü ® * courtyard with a flower bed and a lawn in the middle. Tiles will be placed around the flowerbed. The tapered path to the center of the lawn is also tiled. a Calculate the area of the lawn in m2. Round to the nearest tenth. b Calculate the perimeter of the flower bed in metres. Round to the nearest tenth. c The costs for installing the tiles are €35 per m2. How much does it cost to lay the tiles? Round to the nearest integer. © Noordhoff Uitgevers bv Mixed Exercises 89  Summary Units biz 59 biz 66 biz 81 Examples 44.2 cm = 0.442 m 1335 dm2 = 13.35 m2 3.21 = 3200 cm3 7.1 Length biz 62 The map of the Netherlands has been drawn on a scale of 1:10000000. This means that 1 cm on the map in reality is 10000000cm, or 100 km. A scale of 1:40 means that the dimensions are 40 times larger in reality than the dimensions of the model. 0 100 200 km 7.2 Area biz 63 The perimeter of a figure is the length of its sides. The perimeter of the figure shown on the right is 1.4 + 3 + 2+l + l + l+ 2 + l + 1.4+l = 14.8 cm. The area of the figure is the amount of times that the unit of area fits in the figure. In the figure on the right, 7 whole and 2 half-squares of 1 cm2 fit, making the area 8 cm2. biz 65 area of a rectangle = length width biz 69 To calculate the area of the figure on the right, you divide the figure 12 into rectangles. 10 13------ total area |5 in TII—V = area I + area II + area III = 8 14+ 10 5 + 12 8 The dimensions are in mm. = 112+ 50+ 96 = 258 mm2 90 Chapter 7 Measuring © Noordhoff Uitgevers bv  73 The Circle iz7i.74 circumference = n diameter / diameter = 2 radius n area of a circle = n radius2 3.1H159E65H ✓ X____ For calculations with it you use your calculator. A circle with a 3 cm diameter has circumference = n 3 ~ 9.42 cm radius = 3 - 2 = 1.5 cm area = k 1.52 ~ 7.07 cm2. biz 77 In the figure on the side you see a circle sector with central angle a° coloured in yellow. It follows: area of a circle sector = 777 it radius2 joU length of circle arcAljB = ^^ it diameter è 7.4 Volume biz so The volume of a figure is the amount of times the unit of volume fits in the figure. For example, the volume of the figure on the right is 3 cm3. volume of a cuboid = length width height Three blocks of 1 cm3 each. The volume of the cuboid on the right is 9 2 3 = 54 cm3. biz 82 The surface area of a solid equals the area of its net. A net of the cube on the side consists of two rectangles,with an area of 9 2 = 18 cm2 each. The dimensions are in cm. two rectangles,with an area of 9 3 = 27 cm2 each. two rectangles,with an area of 2 3 = 6 cm2 each. This means the surface area of the cuboid is 2 18 + 2 27 + 2 6 = 102 cm2. 7.5 Views biz84 A front, side and top view show how you look at a figure from three sides. You can measure dimensions in the views. If the scale is known, you can then calculate the actual dimensions. © Noordhoff Uitgevers bv Summary 91  Diagnostic Test 7.1 Length 1 Fill in. i a 7.2 m =... cm c 218 cm =... m e 600 mm =... cm b 0.037 km =... m d 141.1 cm =... mm f 1875 m =... km 2 The Martinitoren in the city of Groningen has a height of 96.8 metres. 2 You can buy models of the Martinitoren in a nearby souvenir shop. The models are on a scale of 1:50. Calculate the height of such a model in cm. 7.2 Area 3 Fill in. 3.4 a 8m2 =... dm2 c 18km2 =... ha e 70 are =... dm2 b 5ha =... m2 d 3000 dm2 =... m2 f 20000 cm2 =... m2 4 A field is 6 by 2.^i. 5 ,6 a A fence will be made around the field. How many metres of fence is needed? b The field is sown with rye. 1.2 kg of seed is needed per are. The seed is available in 25 kg bags. A bag costs € 8.25. Calculate the total price of the rye seed. 5 Calculate the area of the figure on the right. figure 7.60 The dimensions are in cm. 7.3 The Circle 6 In a park lies a round pond with a diameter of s, 9 85 metres. A fence has been placed around the pond. A footpath has also been constructed around the pond. The footpath is 2 metres wide everywhere. See the figure on the right. a Calculate how many metres of fence has been placed. Round to the nearest integer. b Mr Forest walks 1 metre away Trom the pond edge. He covers two kilometres. How many full laps did he make? c Calculate the area of the footpath in m2. Round to the nearest integer. figure 7.61 92 Chapter 7 Measuring © Noordhoff Uitgevers bv  7 The athletics track on the right consists 10, n of straight sections and semicircles. The radius of the inner bend is 36.5 metres. The total length of the track is 400 metres. This is measured at 30 cm from the inner edge of the track. See the red line in the figure. Rounded to integers, the length of a straight section is 84 metres. figure 7.62 a Calculate the length of a straight section in metres, rounded to the nearest tenth. b The central part of the track is a lawn, which is mown twice a week. 120 m2 of grass is mown per minute. How many hours and minutes does it take to mow per week? The javelin sector (starting at Sj is indicated with a ribbon in the inside field. c Calculate how many metres of ribbon has been used. Round to the nearest integer. d Calculate the area of the javelin sector in m2. Round to the nearest integer. 7.4 Volume 8 Fill in. 2, 13, 14 a 1.2 1 =... cl d 3 dm3 =... 1 g 300 cl =... I b 400 dl =... 1 e 0.4 m3 =... dm3 h 3 dm3 =... cm3 c 500 ml =... 1 f 3 dm3 =... cl i 2.5 1 =... ml 9 During a heavy downpour, 44 mm of rain came down in half an hour. is How many litres of water were added to a swimming pool of 50 by 15 metres during that downpour? 10 A box is 2.5 m long, 2.5 m wide and 8 dm high, ie a How many litres is the volume of the box? b How many m2 is the surface area of the box? 7.5 Views 11 See the tent on the right. 17 a Draw the three possible views on a scale of 1:200. b Calculate in m2 the total area of the two sloping roof parts. figure 7.63 The dimensions are in metres. © Noordhoff Uitgevers bv Diagnostic Test 93  Revision 7.1 Length 1 Fill in. a 3 m =... dm g 1 dm =... m 0.001 km b 6 m =... mm h 825 dm =... m 0.01 hm c 1000 cm =... m i 2.7 cm =... mm 0.1 dam d 42 km =... m j 45 mm =... cm 1 m=< 10 dm e 195 m =... km k 400000 cm =... km 100 cm f 0.4 m =... cm I 310 dam =... hm 1000 mm 2 Madurodam is a miniature city in The Hague. Madurodam houses more than 700 models of buildings from all parts of the Netherlands. All the models are made on a scale of 1:25. a Complete the sentence below. At scale 1:25, all the dimensions of the models are... times as... as in reality. b The Dom Tower in Utrecht is 112.5 metres high. How many metres high is the Dom Tower in Madurodam? How many cm is that? c A football stadium has been recreated in Madurodam. The field in the miniature stadium is 42 by 28 dm. What are the dimensions in metres of the football field in reality? Fill in. a 1 m2 =... dm2 e 6 m2 =... cm2 b 8.3 m2 =... dm2 f 0.8 m2 =... cm2 c 720 dm2 =... m2 g 80 cm2 =... mm2 d 800 cm2 =... dm2 h 8000 mm2 =... dm2 94 Chapter 7 Measuring © Noordhoff Uitgevers bv 4 One hectare (ha or hm2) is the area of a square of 1 by 1 hm (1 hm = 100 m). 1 ha = 10 000 m2 1 ha = 100 are 1 km2 = 100 ha 1 are = 100 m2 Fill in. a 6 ha =... m2 d 0.2 ha =... are b 12 km2 =... ha e 7000 are =... ha c 150 ha =... km2 f 2.5 km2 =... ha 5 Of rectangle PQRS, PQ = 15 dm and QR = 7 dm. Remember to include The perimeter of a figure is the sum of the length of the unit in your answer. its edges. a Calculate the perimeter of the rectangle. You calculate the area of a rectangle with area of a rectangle = length width, b Calculate the area of rectangle PQRS. 6 A piece of land is 25 by 18 metres, a There will be a fence around the land. Calculate how many metres of fence is needed. The land is sown with grass. This requires 2.4 kg of grass seed per are. The grass seed is available in 5 kg bags that cost 35 euros each. b How many m2 is the area of the piece of land? How many are is that? c How many kg of grass seed is needed? How many bags are needed? d What is the price for the bags of grass seed in euros? 7 To calculate the area of figure 7.64a, first divide the figure into rectangles. You can see a possible solution in figure 7.64b. You get area = area I + area II + area III. a Calculate the area of the figure in in cm2. figure 7.64 The dimensions are in cm. figure 7.65 The dimensions are in cm. b Calculate in cm2 the area of figure 7.65. © Noordhoff Uitgevers bv Revision 95  73 The Circle 8 Christchurch Park is located in the centre of the English city of Ipswich. In the photo below you can see the pond that lies in this park. The pond is in the shape of a circle. The diameter is 52 metres. You calculate the circumference of a circle with the formula circumference of a circle = n diameter. a Calculate the circumference of the pond in metres. Round to the nearest integer. A footpath has been built around the pond. Mrs Allwood walks on this footpath at a distance of 0.5 metres from the edge of the pond. You calculate the distance she covers in metres when she walks around the pond with 53 zt. b Explain the number 53 in this calculation. c Mrs Allwood walks 1500 metres. Elow many rounds did she walk? 9 The area of a circle is calculated with the formula: area of a circle = n radius2. The circular pond in exercise 8 has a diameter of 52 metres. a How many metres is the radius of the pond? b Calculate the area of the surface of the pond in m2. Round to the nearest tenth. The footpath around the pond is 1.5 metres wide throughout. The pond and the footpath together form a circle. c How many metres is the radius of this circle? d Calculate the area of the circle in m2. Round to the nearest tenth. e You can calculate the area of the footpath by using the answers of questions b and d. Calculate the area of the footpath in m2. Round to the nearest integer. 10 In the figure on the right, you see a rectangle and three semicircles. a How many mm is the diameter of the large circle? And how many mm is the diameter of the small circles? b Calculate the perimeter of the figure in mm. Round to the nearest tenth. c Calculate the area of the figure in mm2. Round to the nearest tenth. 96 Chapter 7 Measuring © Noordhoff Uitgevers bv 11 See the sector with a centre angle of a° on the right. The following applies: area = it radius2 ?60 perimeter = 2 radius + length of arc AB = 2 radius + 22 it diameter 360 In a circle with a radius of 8 cm, a sector with a central angle of 118° is drawn. See the figure on the right, a Calculate the area of the sector in cm2. Round to the nearest integer. b Calculate the perimeter of the sector in cm. Round to the nearest integer. a Fill in. a I m3 =... dm3 c 820 dm3 =... m3 e 84 cm3 =... mm3 b 1.5 hm3 =... dam3 d 3 m3 =... cm3 f 12000 mm3 =... dm3 13 In the diagram on the right, you can see that 1 litre = 10 dl, 1 dl = 10 cl and 1 litre = 100 cl. Fill in. a 50 1 =... dl b 0.3 dl =... cl c 1.41 =... cl d 0.7 1 =... ml e 60 dl =... 1 f 400ml =...l © Noordhoff Uitgevers bv Revision 97 14 In the diagram at exercise 13 on the previous page, you see that 1 litre = 1 dm3, 1 cm3 = 1 ml and 1 dl = 100 cm3. Fill in. a 3 1 =... dm3 c 2 dm3 =... ml e 3000 ml =... dm3 b 5 1 =... cm3 d 1 dm3 =... dl f 90 cl =... dm3 A meadow is 80 metres long and 75 metres wide. a Calculate the area of the meadow. b 40 mm of rain fell on the meadow due to a cloudburst. volume of a cuboid = How many m3 of water has fallen on the Use that 40 mm = 0.04 m. c How many litres is that? 16 The surface area of a solid is the area of its net. On the right you see a cuboid with its net below, a Calculate the surface area of the cuboid, b Calculate the volume of the cuboid. A cuboid is 12 cm long, 8 cm wide and 5 cm high. c Calculate the surface area of the cuboid, d Calculate the volume of the cuboid. 7.5 Views 17 Below you see an object with the right side figure 7.69 The dimensions are in cm. view next to it. 2 figure 7.70 a Draw the front view, the left side view, and the top view, b What view do you need to find the length of edge AB? c Calculate the area of rectangle ABCD. 98 Chapter 7 Measuring © Noordhoff Uitgevers bv  Investigation Letter Arithmetic and Area Some figures can be divided into rectangles. Then you can get the area of the whole figure by adding up the areas of the individual rectangles. In this exercise, you will work with figures constructed from grid squares, and quarter circles inside grid squares. See the figure on the right. We call the area of a grid square x. The area of a quarter circle in a grid square is called y. 1 The area of figure 7.72a is 2x + 3y. a Explain this. You also see three figures with area 2x + 3y. b Why are these three figures not really different? c Draw three possible different figures with area 2x + 3y. This is not allowed. 2 a Draw a figure with area 3x + y, a figure with area x + 3y and a figure with area 4x + 3y. b Draw a figure with area 5x + 4y. Make sure the figure fits into a 3 by 3 square. 3 The area of figure a on the right is x - y. a Explain why. b The area of figure b is 8x - 2y. Explain this. c Give the area of the figures c, d and e. Simplify your answer. © Noordhoff Uitgevers bv Investigation Letter Arithmetic and Area 99  4 Below are five common shapes, which we call basic figures. Write down the area of each basic figure. In the following exercises you may use these areas without explanation. Simplify your answers. 5 Two methods are shown below to find the area of the figure. 100 Chapter 7 Measuring © Noordhoff Uitgevers bv 8 See the figure on the right. a The area of basic figure 1 can be written as x + 2y, but also as 3x - 2z. Explain the second expression with a drawing. b The area of basic figure 1 can also be written as 3y + z. Explain this with a drawing. c Explore how to write the areas of the other four basic figures using the letters x andy, x and z, and y and z. 9 Explore how to write the area of the figure on the right using the letters x and y, x and z, and y and z. © Noordhoff Uitgevers bv Investigation Letter Arithmetic and Area 101