Not for sale or distribution. 6. Measurement. 6.1 Circumference of Circles and Perimeter of Sectors. Get Ready. Exercise 6.

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1 6. Measurement In this chapter you will learn about: calculating the circumference and areas of circles calculating the perimeter and area of sectors calculating the surface area of cylinders calculating the volume of cylinders, prisms, pyramids, and cones calculating the volume of a sphere 6.1 Circumference of Circles and Perimeter of Sectors In this section, you will calculate the circumference of a circle and fi nd the perimeter of sectors. Get Ready You will need: calculator Exercise 6.1 Warm Up 1 Match the names of the parts of a circle written in the cloud to those shown in red: a b c d e circumference sector arc radius diameter 136 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

2 2 a Explaining Explain how you can fi nd the diameter if you know the radius. b Copy and complete this table in your notebook. 3 Use the π button on your calculator to work out: a 2 x π b 4π c 2π x 5 d 12π + 13 Give your answers to 1 d.p. 4 Write down the fraction of each circle that is shaded, cancel the fraction down. The fi rst one is done for you: 30 a = 1 12 b c d Main Exercise 340 Angles at a point add up to 360, so we divide by One formula for fi nding the circumference (C) of a circle is: C = πd, where d = diameter. The diameter of a circle = 2 radius. Rewrite the formula for fi nding the circumference in terms of r. 6 Work out the circumference of the circles. Give each answer to 1 d.p. a b c d 15 cm 2 m Choose which formula you prefer: C = πd or C = 2πr. Make sure you give the correct units. 0.7 cm Radius 123 mm Diameter 3.4 cm?? 12.4 mm? 15 cm 9.8 mm? 124 cm?? m? 137 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

3 7 The circular helipad of a hotel in Dubai has white markings on it: The diameter of the helipad is 33 m. The internal white circle has a radius of 8 m. Work out the total length of the white lines. Round your answer to a sensible degree of accuracy. When rounding an answer, do it at the end of the calculation, and make sure you round it sensibly. How accurately would you need to know the length of a line on a helicopter pad? To the nearest meter, centimeter, millimeter? 8 Afra cuts a semicircle out of cardboard. She wants to decorate the edge with ribbon and work out how much ribbon she needs. a What fraction of the circumference of the circle will she need to calculate? b What other length of ribbon will she need to add? c Copy and complete the calculation: Perimeter of semicircle =? +?? π? d Work out the length of ribbon that Afra needs. 9 Work out the length of the red arc in each diagram. Round your answer to 1 d.p. a b c d cm cm 10 a What is the correct formula for calculating the length of an arc (L) when the angle in the sector is θ? Choose A B, C, or D, and write the formula in your notebook. A L = 1 2 πd B L = θ θ πd C L = πd D L = 360 b Explaining Explain how you know. 11 A clock has a radius measuring 8 cm cm 10 cm Work out what fraction of the circumference you need. 300 πd How far will the end of the second hand travel in: a 1 minute b 30 seconds c 10 seconds In 1 minute the second hand travels the whole way round the clock face cm 138 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

4 12 Explaining Hessa and Eman are calculating the perimeter of a sector. Hessa fi rst works out: on her calculator and rounds it to She then works out: Arc length = 0.06 π 10 = 1.88 cm Perimeter = = cm Eman works out: Perimeter = ( π 10) = cm Who gives the most accurate answer? Explain why. 13 Work out the perimeter of each of these shapes. Give your answers to 1 d.p. a b 30 5 cm 8 m 14 A swimming pool is made in this shape. 20 m 5 m Red tiles are going to be laid around the perimeter. Each red tile is 10 cm long. How many red tiles will be needed? Think about a sensible way to round your answer cm 139 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

5 TALK THINK 6.2 Area of Circles and Sectors In this section, you will calculate the area of a circle and fi nd the area of sectors. Exercise 6.2 Warm Up 1 What fraction of each circle is red? Simplify the answer. a b c d 2 Work out: a 3 2 b 9 2 c π x 2 2 d π x 5 2 e π x a Choose from the cloud all the units of area. mm cm b 2 L What is the same about all the units of area? mm 3 m 2 cm Explain why. cm 3 km km2 ml 4 Fatima is trying to fi nd the area of a circle. g kg mm 2 She cuts a circle up into eight equal sectors and lays them next to one another like this: a What measurement of the circle is approximately equal to the width of the shape? It is made up of?? of the circumference. Fatima cuts the sectors into the smallest pieces she can and lays them next to each other, to make a rectangle. b What measurement of the circle is equal to the width of the shape? c What measurement of the circle is equal to the height of the rectangle? d Write a formula for the area of the rectangle e Fatima can use the fact that the area of her rectangle is the same as the area of her circle. Copy and complete the formula for the area of a circle: A =? width height Multiply the height by the width. 140 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

6 READ Example Work out the area of a circle with radius 7.2 cm. Answer Step 1 Write down the formula for the area of a circle. A = πr 2 Step 2 Substitute the length of the radius into the formula. = π Step 3 Use a calculator to calculate. = π x Step 4 Round the answer, as appropriate. 5 Work out the areas of the circles. Round your answers to 1 d.p. a b c 12 mm 18 cm 4.2 cm = = cm 2 to 1 d.p. 6 Khalid and Ali are calculating the area of a circle. They know the diameter = 10 cm. Khalid writes: Area = πr 2 = π x 10 2 = π x 100 = cm 2 Ali says, I know that you can t be correct because I can draw the circle inside a square with sides of length 10 cm. I know the area of the square = = 100 cm 2, therefore the area of the circle must be less than 100 cm 2. What has Khalid done wrong? Explain what he should do. 7 Copy and complete this table in your notebook. 10 cm 10 cm Radius of Circle Diameter of Circle (Radius) 2 Area of Circle (to 1 d.p.) 1 cm??? 3.1 mm???? 6 mm??? 2.4 cm??? 3.9 m???? 100 cm 2??? 4 cm 2? 10 cm Remember to put the correct units. 141 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

7 THINK 8 The largest possible circle is drawn inside a square with sides of length 12 cm. Work out the area of the circle. Give your answer to 2 d.p. 9 The area of each shape, in terms of r, is written in the cloud. Choose the correct calculation to go with each shape. a b c d e 270 f Calculate the area of these shapes. Give your answer to the nearest whole number. a b c 3 cm m 15 mm 11 The diameter of a circle is 12.5 cm. a Work out the area of the circle to 1 d.p. b A sector with an angle of 45 is cut from the circle. Work out the area of this piece to 1 d.p. 12 Copy and complete this table in your notebook. Draw a diagram to help you. Work out what fraction of the circle you are fi nding the area of. Don t round your answer to part a when you are working out part b. Radius of Sector Angle of Sector Area of Sector (to 1 d.p.) 1 cm 40? 10 cm 60? 120 mm 72? 8 m 25? 3.5 m 140? 1 4πr πr 2 3πr πr 2 3 4πr 2 1 6πr M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

8 13 A circular spinner, with diameter 15 cm is painted in 60 three different colors. 180 io n Work out the area of each colored section to 2 d.p. 14 A clock face is a circle with radius 20 cm. It is painted in 12 different colors ut 12 1 Word fa ct: If two colo rs alterna te they form a pattern, e.g repeating. red, blue, red, blue, red, b lue etc. 5 ib 11 b Another clock is painted in a similar way but this tr a Work out the area of the clock painted yellow. INK TH T IT BOU 15 A pizza restaurant offers two sizes of di s time alternating between two different colors, red and blue. Work out the area painted blue. pizza. sa le or A medium large The medium has a 30 cm diameter, the large has a 45 cm diameter. or Each pizza is cut into 8 slices. Zayed eats two slices of a medium pizza. tf Abdulla eats one slice of a large pizza. Who eats more pizza? Explain A shot put pitch is laid out as shown: 20 m No The angle of the sector is The area inside the pitch is sprayed green. A 1-liter bottle of spray paint covers approximately 30 square meters and costs 60 AED. Work out the approximate cost of spraying this area. Work out the area first. 143 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

9 6.3 Surface Area of a Cylinder In this section, you will calculate the surface area of a cylinder. Get Ready You will need: 3D solids calculator Exercise 6.3 Warm Up 1 a Making Decisions Copy this table into your notebook. Put the units from the cloud into the correct columns. Unit of Length Unit of Area UUnit of Volume??? b Explain how you know which unit is for which measurement. 2 Work out the area of each of these shapes. Make sure you include the units. Give your answers to 2 d.p. where appropriate. a b c d 4 cm 4 cm 6 m 3 What 3D solids are made by these nets? a b c 3 m mm km 3 cm 2 cm mm 3 mm 2 km cm 3 THINK Math fact: The word unit is short for unit of measurement. We must include units otherwise we don t know what the numbers mean imagine being told a line is 3 long! 5 m 15 mm km M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

10 HANDS ON ACtiVITY 4 Making Decisions Ali is calculating the surface area of a cube with sides of length 5 cm. a Which is the correct area of one face? 5 cm 5 cm A 25 cm B 25 cm 2 C 25 cm 3 D 15 cm E 15 cm 2 F 15 cm 3 Explain how you know. b Which is the correct calculation for fi nding the total surface area? A 6 25 B C D 6 (5 + 5) c Which is the correct unit for the surface area of this cube? A cm B cm 2 C cm 3 5 Work out the surface area of: a b c 11 mm 4 cm 7 m Main Exercise 3 cm 7 m 2 mm 2 mm 6 You will need a cylinder. a Draw round the top and bottom of a cylinder. What shapes have you drawn? Look at the curved surface of the cylinder. b What shape would you cut out to make the curved surface of the cylinder? Draw the shape. c Draw a sketch of a net of your cylinder. d Copy and complete in your notebook: The net of a cylinder is made up of two? and one?. 7 m Word fact: A sketch is a diagram not drawn to scale, often with the measurements marked on to describe the size. 5 cm Math fact: The surface area of a shape is found by adding together the area of all the faces (flat surfaces). 145 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

11 7 a Here is a sketch of a cylinder and its net. Draw a sketch of the net in your notebook, and fi ll in the missing information. b Work out the area of each circular piece, to 1 d.p. c Copy and complete in your notebook. The width of the rectangle is equal to the? of the circle. d Work out the area of the rectangle. e Work out the surface area of the cylinder. Add together the areas of both the circles and the rectangle. Give your answers to 1 d.p. 8 Ali uses the formulas for area and circumference of a circle to help him work out a formula for calculating the surface area of a cylinder with any height, h. Copy and complete his working in your notebook. Surface area = (2 area of circle) + (circumference of circle h) = 2 πr 2 + 2πr h =? + 2πrh = 2πr(? + h) 9 Work out the surface area of each of these cylinders. Give your fi nal answers to 1 d.p. a 12 cm b c 18 cm 14 mm 6 mm 21 cm d 5 m 8 cm 3 cm 2 m e 0.5 m? cm? cm 8 m? cm 22 cm Which part of the circle touches this side? Math fact: Area of a circle = πr 2. Circumference of a circle = 2πr. Make sure you use the radius of the circle in your calculations. 146 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

12 HANDS ON ACtiVITY 10 You will need: number cube a Roll a number cube twice. The fi rst number is the height of the cylinder. The second number is the radius of the circular face. Work out the surface area of a cylinder with these measurements. b Swap the two numbers around and work out the new surface area. c Repeat three times. d When are your answers to parts a and b going to be the same? 11 A cardboard tube has no top or bottom. Work out the area of cardboard you need to make it. Give your answer to 2 d.p. 2 cm You only need to fi nd the area of the rectangular section. 12 A glass is made in the shape of a cylinder. It is 12cm tall and has a base with diameter 8 cm Work out the surface area of the outside of the glass to 1 d.p. 13 A cylindrical pipe is 23 m long. The radius of each end is 0.4 m. Work out the surface area of the pipe. Give your answer to an appropriate degree of accuracy. Word fact: cylindrical means a shape that is a cylinder. 30 cm 147 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

13 Example Work out the surface area of this shape: Answer Step 1 Think about which sides you should include. Are there any you should subtract? The area of the base of the cylinder cannot be seen, and the same amount of the cuboid is not needed, so we can subtract both circular faces of the cylinder. Total surface area = surface area of cuboid + area of curved side of the cylinder. Step 2 Work out the surface area of Area of front of rectangle = 24 4 each different face of the shape. = 9 2 Area of side of rectangle = 3 4 = 12 cm 2 Area of top of rectangle = 24 3 = 72 cm 2 Area of curved side of cylinder = 2 π 6 2 Step 3 Add together the areas you need. Simplify the calculation if you can. (The top and bottom of the cylinders are the same, the front and back are the same and the sides are the same. Surface area = 2 (front) + 2(side) + 2 (top) + curved side of cylinder = (2 96) + (2 12) + (2 72) = = cm 2 14 For each part of this question make models of the shapes using 3D solids, so you can see which faces need to be subtracted to get the fi nal surface area. Then work out the surface area of these shapes, using the given measurements. Give your answer to 1 d.p. where appropriate. a 2 mm 4 cm 5 mm 9 mm 4 mm 4 mm b 12 cm 10 cm 24 cm 2 cm 4 cm 3 cm Always write which area you are calculating, so you know what to add together, and don t forget the units. Think about the area you would have to paint if you were painting the shape. 148 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

14 THINK TALK 15 This solid cylinder has a hole through the middle, with a diameter of 11 cm. Latifa covers the outside of the cylinder with blue paper. a Work out the area of a circle with diameter 25 cm. diameter of cylinder = 25 cm b Work out the area of a circle with diameter 11 cm. c Use your answers to parts a and b to work out the area of the end of the cylinder. d Work out the area of the curved side of the cylinder. e Use your answers to parts c and d to work out the area Latifa will cover with paper. 16 Drainpipes are made in 3 m long pieces with a diameter of 8.8 cm. What is the surface area of one piece of drainpipe? Give your answer in square meters to the nearest Change all measurements to 0.01 m 2. meters before you do the calculation. 17 A polytunnel is made up of a half cylinder. Each polytunnel is 15 m long and has a radius of 50 cm. Work out the area of plastic used to make one polytunnel. Give your answer to the nearest square meter. 18 Generating Solutions A greenhouse is made of a half cylinder and a rectangular prism. It needs a new shade cloth to cover it completely. The cloth cost 75 AED per m 2. Work out the cost of the shade cloth. 10 m Work out the surface area of the shape fi rst, remember it won t need cloth on the fl oor of the greenhouse. 25 cm diameter of hole = 11 cm 19 For some questions in this exercise the formula for the surface area of a cylinder was useful. For others is wasn t. What advice would you give to another student about using the formula for the surface area of a cylinder? 3 m 1.5 m 1 m 149 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

15 TALK 6.4 Volume of 3D Shapes In this section, you will calculate the volume of prisms, cylinders, pyramids, and cones. Get Ready You will need: 3D solids calculator Exercise 6.4 Warm Up 1 Choose the correct shape from the cloud to complete these formulas. Area of a? = 1 2 (a + b) h Area of a? = 1 2 b h Area of a? = πr 2 Area of a? = b h Volume of a? = l b h 2 Work out the area of these shapes. a b c d e 5 cm 3 cm 4 cm 5 mm 5 cm 3 cm 8 m 3 Jasem is putting cm 3 blocks into rectangular prisms. a How many cm 3 can he fi t into each shape? i ii iii 12 cm 10 mm 150 mm 3 cm 2 cm b Explain how you could work this out without counting. f 5 cm 7 m 5 cm 5 cm 5 cm 4 cm 2 cm parallelogram triangle circle rectangular prism 12 mm 9 cm 1 cm trapezium Make sure you give the units. 1 cm 3 1 cm 3 1 cm 3 c Write down the volume of each of the shapes in part a. 150 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

16 Math fact: HANDS ON ACtiVITY 4 Work in pairs. Look at a collection of 3D shapes. Find all the shapes that are prisms. For each one, decide which faces are the identical opposite parallel end faces. Main Exercise 5 a Work out the area of the circular end of the prism. b Copy and complete the calculation in your notebook: Volume = 21.? =? cm 3 21 cm 15 cm Give your answers to 2 d.p. 6 Explaining If r = radius, d = diameter and h = height of a cylinder which is the formula for calculating the volume of a cylinder? Explain why. a πrh b πr 2 h c πdh d πd 2 h 7 Making Decisions Match each cylinder to its volume. Cylinder Volume A i 18.8 cm 3 B ii 2.4 cm 3 1 cm C iii cm D iv cm 2 cm 3 cm Area = 28.3 cm E 2 v 25.1 cm 3 F Area = 3.14 cm 2 vi cm 1 cm 2 cm A prism is a 3D shape which has two identical opposite parallel end faces. This shape is a prism and the red triangles are the identical opposite parallel end faces. Math fact: The volume of a prism is found by multiplying the area of one of the identical parallel faces (the end face) by the length of the prism. 151 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

17 8 K TAOL UT IT Ibrahim says, I could work out the answers to question 6 without using a calculator. Example io n Explain how he could do this. 20 cm 1 Work out the volume of this prism: Answer 10 cm 12 cm Step 1 Write down the formula for the volume of a prism. ut AB ib Volume = area of end face x length Step 2 Substitute the measurements in the formula. 1 tr = ( ) 10 Step 3 Calculate and put in the units. = Work out the volume of these prisms: a b 7.5 cm 8 cm Area = 9.2 cm2 7 cm c 12 cm d Area = 15 mm2 3 cm or 9 di s = 960 cm3 23 mm 2 cm 11 cm 5 cm sa le If you are given the area of the end, just multiply this by the length of the prism to find the volume. 10 You will need: nets of a cone, a cylinder, a square-based pyramid, and a rectangular prism sticky tape tf or N DS O HAANCtiVITY thin card sand or rice for pouring No a Make the cone, from the net. Carefully stick the overlap down. Make the cylinder. Word fac Check the cylinder and cone have the same height and same diameter. You should have: cone cylinder t: The vertic al height of a cone, cylinder, or pyramid is how tall it stands off the ground. 152 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

18 HANDS ON ACtiVITY b Fill the cone with the sand/rice and pour it into the cylinder. How many times can you do this before the cylinder is full? c Copy and complete in your notebook: Volume of cylinder =? volume of cone. Volume of cone =? volume of cylinder. d Repeat parts a c but use the nets for the square-based pyramid and rectangular prism with the same base and height as the pyramid. e Copy and complete in your notebook: square-based pyramid Volume of prism =? volume of pyramid. Volume of pyramid =? volume of prism. 11 Work out the volume of each of these pyramids. a b c m What fraction of a cylinder is a cone? 9 m 12 m You will need: number cube Roll the number cube twice to give you values for a and b in the cone. a Work out the volume of the cone using these values of a and b. Remember the volume of a cone is 1 3 A h, and the area of the circular base is πr 2. b Repeat three times. c What is the: 3 mm 12 mm 15 mm 1 18 cm i largest volume you could fi nd? ii smallest volume you could fi nd? rectangular prism 4 cm Math fact: The volume of a pyramid or cone with the same base and height as a prism or cylinder is: Volume = 1 3 A h, where A = area of base, and h = vertical height of pyramid/cone. a b 153 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

19 THINK 13 Ahmed is calculating the height of a cone. He knows the volume = 100 cm 3 and the radius of the base = 5 cm. Copy and complete the calculation to work out the height of the cone: 1 3 (π r2 ) x h = Volume 1 3 (π 52 ) x h = π h = 100? h = cm h =?. Give your answer to 1 d.p. 14 Use Ahmed s method to work out the height of a cone if: a Volume = 300 cm 3, radius of base = 12 cm. b Volume = cm 3, radius of base = 5 cm. c Volume = 250 cm 3, diameter of base = 4 cm. Give your answers to 1 d.p. 7 cm 15 A paper cup is made in the shape of a cone. The manufacturer wants it to hold 300 ml of water. If the diameter of the cone is 7 cm what is the 1 ml = 1 cm 3 height of the cone? Give your answer to the nearest 0.1 cm. 16 A perfume bottle is designed in the shape of a rectangular prism with a pyramid on the top. The rectangular prism is 8 cm tall. Will the bottle hold 150 ml of perfume? Work out the volume of each shape separately. 4 cm 4 cm 12 cm 154 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

20 6.5 Volume of Spheres In this section, you will calculate the volume of spheres. Get Ready You will need: sphere calculator Exercise 6.5 Warm Up 1 Work out: a 4 3 b 1 3 c 5 3 d Making Decisions Decide which of the statements are true and which are false. a A sphere is not symmetrical. b A sphere has one face. c A sphere has one vertex. d All the points on the surface of a sphere e are the same distance from the center. 3 Copy these sentences into your notebook. Fill in the spaces using words from the cloud. Word fact: The vertex of a shape is the corner. A cube has 8 vertices. A sphere is a? dimensional shape with? face and no.? If you cut a sphere into? pieces through the center, the two pieces will be.? The radius of a? is the distance from the center to the?. Math fact: A solid round figure is called a sphere. A ball is a sphere. three identical two vertices sphere one outside 155 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

21 THINK READ THINK Main Exercise 4 The radius of a sphere is 3 cm. Maitha says, I can estimate the volume of the sphere by putting the sphere in the smallest cube it will fi t into and fi nding the volume of that. a What would Maitha get as the approximate volume of the sphere? Khawla says, I can get an even better estimate by putting the sphere in the smallest cylinder possible and fi nding the volume of that. b What would Khawla get as her approximate volume of the sphere? c Are their answers too big or too small? d Explaining Who has the closest answer? Explain why. 5 Maitha and Khawla design an experiment to find a more accurate value for the volume of a sphere with radius 3 cm. They use a sphere and a cylinder with the diameter and height both equal to. They fi ll the sphere with water, and then pour this into the cylinder. They then measure the height the water reaches in the cylinder. Khawla says, This shows that the volume of the sphere is less than the volume of the cylinder, because of the space around it. The height of the cylinder is, but the water only reaches 4 cm. The missing 2 cm is the space around the sphere inside the cylinder. The water from the sphere fi lls a fraction of the cylinder. a What fraction of the cylinder is fi lled? b Maitha says, I think the volume of a sphere will always be the same fraction of the volume of a cylinder with an equal radius, so we can write a formula. Copy and complete her working. For a sphere with radius r, the cylinder will have a radius of r and a height of 2r. 3 cm Write the formula. volume of the cylinder = πr 2 h, 3 cm Substitute 2r for h. volume of cylinder = πr 2? Simplify: =.? Volume of sphere =?? cylinder =?? 2πr 3 4 cm Height of water in the cylinder/height of the cylinder when full. Multiply the number and fraction together. =?? πr M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

22 6 The radius of a sphere is 5 cm. Copy and complete the calculation in your notebook, using Maitha s formula. Volume =?? πr 3 =?? π 3? =? cm 3 Give your answer to 2 d.p. 7 Work out the volume of the spheres, give your answer to 1 d.p. a b c 9 cm d e f 15 cm 8 Ali estimates one of the domes on the Sheikh Zayed Mosque has a diameter of 32 m. Although the dome is not a whole sphere, Ali estimates the volume by working out 4 5 of the volume of a sphere. What would Ali s estimate be for the volume of the dome, to the nearest cubic meter? 9 A hemisphere is put on top of a cylinder. Work out the volume of the shape to 1 d.p. Work out the volume of the hemisphere and cylinder separately. 10 A plastic stand is made for ice cream cones. 10 cm 10 cm 14 mm 10 cm 11 m 30 cm Work out the volume of plastic used to the nearest cm 3. 2 m 3.2 cm 14 cm Math fact: Volume of sphere = 4 3 πr 3, where r =radius of sphere. Word fact: A hemisphere is half of a sphere. r 157 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

23 THINK THINK tennis balls are packed into a cubic box with sides of length 20 cm. The balls have a diameter of 6.. How much space inside the box is not used? Give your answer to 2 d.p. 12 A sphere and a cube have equal volume. 20 cm Given that the cube has sides of length 10 cm, what is the radius of the sphere to the nearest whole number? 13 Faisal owns an ice cream shop. 6. He buys ice cream in 5-liter boxes which cost him 100 AED. For each ice cream he puts two spherical scoops into a cone or pot. The ice cream scoop has a radius of 3 cm. He sells each ice cream for 10 AED. a How many ice creams can Faisal make from the 5-liter box? b How much profi t does he make if he sells all 5 liters? c Comment on how accurate you think Khalid s estimate might be. 5 liters = ml = cm 3 20 cm 20 cm 158 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

24 Summary The circumference of a circle = 2πr The length of an arc is a fraction of the circumference, and is calculated using the formula: Arc Length = θ θ 360 πd or = 360 2πr The area of a circle = πr 2 The area of a sector is a fraction of the area of a circle, and is calculated using the formula: Area of Sector = θ 360 πr 2 The surface area of a cylinder is calculated by considering the net, made up of two circles and a rectangle with length equal to the circumference of the circles. Surface Area of a cylinder = 2πr 2 + 2πrh = 2πr (r + h) The volume of a prism = area of face length area r r r h h length Volume of a cylinder = πr 2 h Volume of a cone = 1 3 πr 2 h Volume of a pyramid = 1 3 volume of prism with same face and same height Volume of a sphere = 4 3 πr cm 60 volume = volume of 159 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

25 THINK Review Use a calculator, and give all your answers to 1 d.p. unless instructed otherwise. 1 Work out the circumference and area of the circles with: a radius = 3.2 cm b diameter = 9.5 m 2 Calculate the perimeter and area of this shape: What is the angle of the sector? 3 Generating Solutions A garden is in the shape of a sector of a circle. The angle in the sector is 75 and the length of one of the straight sides is 23.5 meters. The garden is fenced around the outside. The fencing costs 50 AED per meter. The garden is also going to be covered in grass. Each box of grass seed covers 15m 2 and costs 75 AED. How much will it cost altogether to fence and buy grass seed for the garden? Draw a sketch of the garden. (Give your answer to the nearest dirham). 4 A cardboard tube is made to hold sweets. 4 cm One end is cardboard and the lid is plastic. The diameter of the tube is 4 cm and the length of the tube is 15 cm. What area of cardboard is needed for the tube? 15 cm Which areas will you need to add together? 5 Organizing Information Work out the volume of each of the three shapes. Put them in order of size from smallest to largest. a 2 cm b 9.0 m c 11 cm 4.0 mm 2.0 m 2.5 mm 5.0 m 6.2 mm 6 A cylindrical pipe has a radius of 4 cm and length of 2 m. How much liquid will it hold? Calculate the volume in cm 3, and give your answer in liters. 7 An ice cream cone is 14 cm tall and the top has a diameter of 8 cm. The ice cream melts, what is the maximum volume of melted ice cream it can hold? 8 A football has a diameter of 22 cm (to the nearest cm). Work out the volume of the ball. Assume the ball is a sphere. 3 cm mm 160 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

26 Put it all together 9 Organizing Information Match the formulas to the correct descriptions and copy them into your notebook. 1 A Volume of a sphere 3 πr 2 h θ B Volume of a cone 360 πd θ C Volume of a cylinder 360 πr 2 D Surface area of a cylinder πr 2 E Area of a sector 2 πr(r + h) F Length of an arc 4 3 πr 3 G Area of a circle πr 2 h H Circumference of a circle 2πr 10 A wooden box is made in the shape of a half cylinder as the lid with a rectangular prism as the base. a Work out the volume of the box. Varnish costs 50 AED for a small can that will cover up to cm 2. b The box needs two coats of varnish, will three cans be enough? 11 The front face of a prism is an equilateral triangle with sides of 2 cm. If the prism is 10 cm long what is the volume of the prism? 2 cm 2 cm 2 cm 10 cm 8 cm 1 24 cm Use Pythagoras Theorem to fi nd the height of the triangle. 161 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

27 Put it all together THINK 12 Three different solids are made using these two pieces: a 2a Match the solids A, B, and C to their surface areas: shape A i 4.5πa 2 ii 2πa a 2 iii 2πa a 2 Substitute the letters into the formulas for surface area, and simplify the expressions. I can statements What can you do? I can fi nd the circumference of a circle. I can fi nd the length of an arc. I can fi nd the area of a circle. I can fi nd the area of a sector. I can fi nd the surface area of a cylinder. I can fi nd the volume of a prism. I can fi nd the volume of a cylinder. I can fi nd the volume of a sphere. a a shape B 2a shape C 162 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30

28 163 M06_ADEC_SB_09_ARW_8265_U06.indd /04/ :30