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# Maths A - Chapter 4

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### Maths A - Chapter 4

1. 1. syllabussyllabusrrefefererenceence Strand: Applied geometry Core topic: Elements of applied geometry In thisIn this chachapterpter 4A Changing units and calculating perimeters 4B Calculating areas 4C Total surface area 4D Volume and capacity 4 Length, area and volume MQ Maths A Yr 11 - 04 Page 101 Wednesday, July 4, 2001 4:11 PM
2. 2. 102 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Introduction Peta is an expert in the craft of origami (the Japanese art of paper folding). She deftly folds and creases paper, the end product being a crane, an elegant swan, a gift box . . . the list is endless. In fact, many two-dimensional and three-dimensional objects can be folded from a single piece of paper without any cutting. In creating these works of art, it is necessary to have an understanding of the spatial relationships of geometry. It is essential to fold the paper at the correct angles and to have the side lengths in the correct proportions. We are surrounded by geometric shapes — in our daily lives, in nature, architecture, etc. Throughout this chapter you’ll discover prop- erties of geometry by conducting investi- gations. Each investigation is designed to reinforce the particular geometry concept of that section of study. MQ Maths A Yr 11 - 04 Page 102 Wednesday, July 4, 2001 4:11 PM
3. 3. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 103 1 What do you understand by the word length? 2 Give some units you would use to measure length. 3 What do you understand by the term perimeter? 4 Calculate the perimeter of the following shapes: a b c d 5 Deﬁne the term area. 6 Give some units you would use to measure area. 7 Calculate the areas of the ﬁgures in question 4 above. 8 Identify the shapes of the following ﬁgures: a b c d e f 9 Use your calculator to determine answers to the following (correct to 2 decimal places, if necessary). a 1.462 b c 4.33 d 10 Distinguish between the terms volume and capacity. 11 Give some units you could use to measure a volume b capacity. 12 Convert the following quantities to the units indicated. a 6 mm → cm b 0.25 m → mm c 40 cm2 → m2 d 0.45 km → m e 300 mL → L f 5000 cm3 → m3 g 25 L → kL h 5 cm3 → mm3 i 100 cm3 → L j 25 kL → m3 Objects all around us can be classiﬁed as two-dimensional (2-D) or three-dimensional (3-D). A two-dimensional object can be described by two measurements (for example a length and a width), whereas a three-dimensional object requires three measurements to describe its shape (for example a length, a width and a height). 6 cm 10 m 4 m 4.33 m 5 m 3 m SkillS HEET 4.1 E XCEL Spread sheet Rounding 3.07 28.763 MQ Maths A Yr 11 - 04 Page 103 Wednesday, July 4, 2001 4:11 PM
4. 4. 104 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d It’s important to realise that sometimes it is not possible to construct a 3-D object that has been drawn on paper in two dimensions. Look carefully and you will see that this is the case in the situation at right. Try drawing the shape and you will understand the problem! Paper folding 1 Constructing a 3-D package from a 2-D shape Resources: Cardboard, scissors, protractor, ruler. For our ﬁrst investigation, let’s look at the relative positions, shapes and sizes of the resulting faces when a 2-D piece of cardboard is folded into a 3-D shape to form an unusual postage box. 1 Trace the above shape onto a piece of thin cardboard. Basically, it consists of two rectangles. 2 Cut along all the blue lines. 3 The lines marked – – – – represent ‘valley’ folds. Crease ﬁrmly along these lines with the fold pointing down like a valley. 4 The line XY marked – – • – – • represents a ‘mountain’ fold. Crease ﬁrmly along this line, with the fold pointing upwards like a mountain. 5 Rotate the top rectangle through 90º at the point Y, along the crease line XY, in an anticlockwise direction, until it lies on top of the lower rectangle. 6 Fold the top and bottom ﬂaps in along the crease lines in a 3-D manner. Fold in the side ﬂaps along the crease lines in a 3-D manner. 7 This container, on a larger scale, would be a suitable package for posting rectangular objects through the mail. 8 Open the package and a measure angle ZXY b measure angle a. 9 Would it be possible to vary these angles and still form a rectangular container? Investigate the consequences. inv estigat ioninv estigat ion w2 w2 w1w1 a X Z Y MQ Maths A Yr 11 - 04 Page 104 Wednesday, July 4, 2001 4:11 PM
5. 5. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 105 Two-dimensional objects As mentioned previously, the shape of 2-D objects can be speciﬁed by two measure- ments. These objects lie in one plane. Among other properties, they possess a perimeter and an area. Perimeter The perimeter represents the distance around the boundary of a ﬁgure. (We are assuming that all the ﬁgures we are dealing with are closed; that is, they begin and end at the same point.) Any line inside the boundary is ignored when calculating the perim- eter. The units used to measure perimeter are those of linear measure: millimetre (mm), centimetre (cm), metre (m) and kilometre (km). We’ll ﬁrst discuss converting measurements from one unit to another. You are fami- liar with the following conversions. 10 millimetres = 1 centimetre 100 centimetres = 1 metre 1000 metres = 1 kilometre Using these conversion factors, we can now construct a ‘conversion ladder’. The smallest unit (mm) is at the narrow top of the ladder; working down the rungs we approach the largest unit (km) which is at the wide base of the ladder. The con- version factors are placed on the rungs in between the units. In changing measurements from a smaller unit to a larger unit, we divide by the relevant conversion factor(s) because we know that our answer must be a lesser amount. When converting from a larger to a smaller unit, we multiply by the relevant conversion factor(s) since we know that we require a greater amount as the result. 10 Consider using this container to post a book. On your ﬂat shape, identify: a on which part the book would lie b which sections would dictate the maximum length, width and thickness of the book. 11 Why is the width w1 greater than the width w2? 12 Taking the measurements of your text book, design and construct a package of this style which could be used to post your book. SkillS HEET 4.2 mm 10 cm 100 m 1000 km Multiply when going to a smaller unit Divide when going to a larger unit Smallest unit × ÷ Largest unit MQ Maths A Yr 11 - 04 Page 105 Wednesday, July 4, 2001 4:11 PM
6. 6. 106 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d We shall now review familiar formulas used to determine the perimeter of common shapes. Shape Perimeter Square P = 4S Rectangle P = 2(L + W) Other polygons P = sum of lengths of all sides Circle C = 2πr or C = πD Sector C = length of arc + 2 radii C = × 2πr + 2r Complete each of the following. a 30 mm = cm b 4800 m = km c 6.5 m = cm d 8400 mm = m THINK WRITE a Changing millimetres to centimetres: divide by 10. a 30 mm = 30 ÷ 10 cm = 3 cm b To change metres to kilometres: divide by 1000. b 4800 m = 4800 ÷ 1000 km = 4.8 km c To change metres to centimetres: multiply by 100. c 6.5 m = 6.5 × 100 cm = 650 cm d To change millimetres to metres: divide by 10 (to change to centimetres) then divide by 100 (to change to metres). d 8400 mm = 8400 ÷ 10 cm = 840 ÷ 100 m = 8.4 m 1WORKEDExample EXCE L Spreadshe et Length con- versions Mat hcad Unit con- versions SkillS HEET 4.3 SkillS HEET 4.4 S W L r D Arc length Arc θ° r θ° 360° ----------- MQ Maths A Yr 11 - 04 Page 106 Wednesday, July 4, 2001 4:11 PM
7. 7. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 107 Perimeter of composite ﬁgures In many instances, ﬁgures are not of one distinct shape; they may be composed of sev- eral shapes. The perimeter of such shapes is still the distance around the boundary of the composite ﬁgure. Remember to ignore any lines inside the ﬁgure. It is often helpful to start at one point in the ﬁgure, work your way around the boundary in a clockwise or anticlockwise direction, identifying shapes and adding the lengths of all sides, until you reach your starting point. Find the perimeter around the following piece of pizza. THINK WRITE Identify the shape: sector. Perimeter means ‘distance around outside’; that is, arc plus 2 radii. Write the appropriate formula. P = × 2πr + 2r Identify the values of the variables. θ° = 40°, r = 10 cm Substitute values in the formula. Calculate the answer, not forgetting units. P = + (2 × 10) P = 6.98 + 20 P = 26.98 cm 10 cm 40° 1 2 3 θ° 360° ----------- 4 5 6 40 360 --------- 2× π× 10×     2WORKEDExample Find the perimeter of this shape. THINK WRITE Start at X and travel in a clockwise direction until reaching X again. Identify the sides as 3 straight lines and one semicircle. P = Side 1 + Side 2 + Side 3 + circumference of circle Write the formulas. P = S1 + S2 + S3 + πD Identify the values of the variables. S1 = 20, S2 = 10, S3 = 20, D = 10 Substitute the values of the variables in the formula. Calculate the answer, not forgetting units. P = 20 + 10 + 20 + ( × π × 10) P = 50 + 15.7 P = 65.7 cm X20 cm 10 cm 1 1 2 --- 2 1 2 --- 3 4 5 1 2 --- 3WORKEDExample MQ Maths A Yr 11 - 04 Page 107 Wednesday, July 4, 2001 4:11 PM
8. 8. 108 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Changing units and calculating perimeters 1 Copy and complete each of the following. 2 Richard is planning to have a garage built. The garage is 5.2 m long, 2.4 m wide and 2.5 m high. All builders, however, work in millimetres. What are the dimensions of the garage, in millimetres? 3 Find the perimeters of the following ﬁgures (to the nearest whole units). a b c d e f 4 Find the perimeters of the following ﬁgures. a b c a 70 mm = cm b 600 cm = m c 5000 m = km d 9 cm = mm e 12 m = cm f 9 km = m g 86 mm = cm h 9.2 km = m i 2400 m = km j 6.4 cm = mm k 11.25 m = cm l 2.2 cm = mm remember 1. Recall unit conversions for length. 2. Multiply when changing to a smaller unit and divide when changing to a larger unit. 3. The perimeter is the distance around a closed, 2-dimensional ﬁgure. 4. When ﬁnding the perimeter of a composite ﬁgure, start at any place on the perimeter and continue in a clockwise or anticlockwise direction until reaching the starting point. Ignore any lines inside the perimeter. 5. Don’t forget to include units in the answer. remember 4A WWORKED Example 1 WORKED Example 2 7m 12 m 5 m 4 m 23.7 cm 17.8cm 15.4cm 27.5 cm 13.5 mm 7.5 m 11.5 m 5 m 4m 120m 210m 90 m WORKED Example 3 12 m 25 m 10 m 14 m 20 m 2 m 3.5m 12 m 17m MQ Maths A Yr 11 - 04 Page 108 Wednesday, July 4, 2001 4:11 PM
9. 9. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 109 d e f 5 The perimeter of the ﬁgure shown in centimetres is: A 34 B 24 + 5π C 24 + 2.5π D 29 + 5π E 29 + 2.5π 6 The perimeter of the enclosed ﬁgure shown is 156.6 metres. The unknown length, x, is closest to: A 20.5 m B 35.2 m C 40.2 m D 80.4 m E Cannot be determined Paper folding 2 Resources: A4 paper, scissors. Having looked at some 2-D shapes, let’s investigate a ‘Geometry jigsaw’. 1 Cut a square from a sheet of unlined A4 paper. Scribble a pattern on one side of the paper, so that you can distinguish the ‘top’ of the jigsaw from the ‘underneath’. 2 Fold the square sheet in half along its diagonal. Unfold it, and cut it along the crease. You should now have two triangles. 3 Take one of these triangles, fold it in half and cut it along the crease line. 4 Take your second triangle from above and lightly crease it to ﬁnd the midpoint of the longest side. Fold it so that the vertex of the right angle touches that midpoint and cut it along the crease. This forms a trapezium and a small triangle. 5 Take the trapezium, fold it in half and cut along the crease line. You should now have two smaller trapeziums. 125 mm 24 mm 90 mm48 mm 21 cm 16 cm 8cm 10cm 12 cm 10 cm 20 m 22 m 7 m 13m 11m 34m 44m mmultiple choiceultiple choice 12 cm 7 cm 2 cm 3 cm mmultiple choiceultiple choice 35.2m 20.5 m x inv estigat ioninv estigat ion MQ Maths A Yr 11 - 04 Page 109 Wednesday, July 4, 2001 4:11 PM
10. 10. 110 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 6 Fold the acute base angle of one of the trapeziums to the adjacent right base angle and cut along the crease. This should result in a square and a small triangle. 7 Take your other trapezium and fold its right base angle to the opposite obtuse angle. Cut along the crease. This should result in a parallelogram and a small triangle. 8 You should now have 7 shapes: 5 triangles, 1 square and 1 parallelogram. 9 See if you can now assemble this Geometry jigsaw to form the original square from which it was cut. 10 See how many different shapes you can form using some or all of the 7 shapes. This investigation should enhance your understanding of geometric shapes. Task 1 Take your 7 shapes. Use the smallest triangle as the basic unit of area. Arrange your pieces in order of increasing area. Give the area of each piece in terms of ‘small triangular units’. Some shapes may have the same area. Copy and complete the following table: Task 2 Your pieces can be ﬁtted together in different combinations to form squares of vari- ous sizes. Experiment with your pieces to see whether you can form a square using the following number of pieces. Copy and complete the table below. Shape Sketch of shape Area (e.g. 3 ‘triangular units’) 1 2 3 4 5 6 7 Pieces Sketch Pieces Sketch 1 5 2 6 3 7 4 MQ Maths A Yr 11 - 04 Page 110 Wednesday, July 4, 2001 4:11 PM
11. 11. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 111 Area Area represents the amount of space within the boundary of a closed ﬁgure. The units used to measure area are those of square measure: mm2 , cm2 , m2 and km2 . There are two units of area which are not square units — the hectare (ha) in the metric system and the acre in the imperial system. By constructing a conversion ladder, as we did before for linear measure, we ﬁnd: To obtain the conversion factors for square measures, it is necessary to square the linear measure conversion factors. As previously, we multiply when converting to a smaller unit and divide when converting to a larger unit. One hectare is equivalent to the area of a square of side length 100 metres. So 1 ha = 100 m × 100 m That is 1 ha = 10 000 m2 Square measure Resources: Paper, pencil, ruler. How is it possible to determine unit conversions for square measure knowing the relevant conversion for linear measure? 1 Draw a square of side 1 cm. 2 You are aware that each centimetre can be divided into 10 millimetres. Mark these millimetre divisions on each side of the square, then join opposite side markings. 3 This creates a grid of smaller squares. How many of these smaller squares are there? Each smaller square is 1 mm2 while the large square is 1 cm2 . So how many mm2 are there in 1 cm2 ? 4 Use the reasoning applied above (it is not really practicable to draw a square of side length 1 metre) to determine the number of cm2 in 1 m2 . Converting units in square measure is simply a matter of applying the above technique. inv estigat ioninv estigat ion mm2 10 × 10 = 100 cm2 100 × 100 = 10 000 m2 1000 × 1000 = 1000 000 km2 × ÷ Math cad Unit conversions 100 m 100 m1 ha MQ Maths A Yr 11 - 04 Page 111 Wednesday, July 4, 2001 4:11 PM
12. 12. 112 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d We’ll now review familiar formulas used to ﬁnd the areas of common shapes. Shape Area Square A = S2 Rectangle A = L × W Parallelogram A = b × h where the height measurement must be at a right angle to the base measurement. Trapezium A = (a + b)h where the height measurement must be at a right angle to the base measurement. Triangle A = bh where the height measurement must be at a right angle to the base measurement. Complete each of the following: a 250 mm2 = _____ cm2 b 5 km2 = _____ ha THINK WRITE a Change mm2 to cm2 , so divide by 100. a 250 mm2 = 250 ÷ 100 cm2 250 mm2 = 2.5 cm2 b Change km2 to m2 (multiply by 1 000 000). Change m2 to ha (divide by 10 000). b 5 km2 = 5 × 1 000 000 m2 5 km2 = 5 000 000 m2 5 km2 = 5 000 000 ÷ 10 000 ha 5 km2 = 500 ha 4WORKEDExample EXCE L Spreadshe et Area of a square S EXCE L Spreadshe et Area of a rectangle L W Area of a parallelogram b h b h a 1 2 --- Area of a triangle Vic Gen fig ch 6. 57a b h G h b h b 1 2 --- MQ Maths A Yr 11 - 04 Page 112 Wednesday, July 4, 2001 4:11 PM
13. 13. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 113 Triangle (Heron’s formula) A = where s = (a + b + c) (Use when height measurement is unknown.) Circle A = π r2 Sector A = × πr2 Shape Area a bc s s a–( ) s b–( ) s c–( ) 1 2 --- E XCEL Spread sheet Area of a circle r D Cabri Geo metry Area of a circler θ° θ° 360° ----------- Find the area of the triangle at right. THINK WRITE Write the formula. A = × b × h Substitute for the base and the height. = × 12.8 × 9.4 Calculate the area. = 60.16 cm2 9.4 cm 12.8 cm1 1 2 --- 2 1 2 --- 3 5WORKEDExample Find the area of each of the following shapes. a b THINK WRITE a Write the formula. a A = b × h Substitute the base and height. A = 14 × 9 Calculate the area. A = 126 m2 b Write the formula. b A = × (a + b) × h Substitute the sides and height. A = × (5.9 + 11.4) × 7.2 Calculate the area. A = 62.28 cm2 9 m 14 m 7.2 cm 5.9 cm 11.4 cm 1 2 3 1 1 2 --- 2 1 2 --- 3 6WORKEDExample E XCEL Spread sheet Area of a triangle Math cad Substitu- tion 2 Cabri Geo metry Area of a trapezium MQ Maths A Yr 11 - 04 Page 113 Wednesday, July 4, 2001 4:11 PM
14. 14. 114 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Mat hcad Area of a triangle Find the area of this shape. THINK WRITE Identify the shape (in this case it is a triangle with no height measurement) and write down the appropriate formula for the area (Heron’s formula). A = where s = (a + b + c) Identify the values of the pronumerals. a = 7, b = 8, c = 10 To ﬁnd s, substitute a, b and c values into the formula and simplify. s = (7 + 8 + 10) = × 25 = 12.5 Substitute the values of a, b, c, and s into the formula for the area. A = Simplify. (a) Evaluate the brackets ﬁrst. (b) Multiply the values together. (c) Take the square root. A = = = 27.810 744 33 (Round off the answer to 1 decimal place and include the units.) A = 27.8 cm2 7 m 8 m 10 m1 s s a–( ) s b–( ) s c–( ) 1 2 --- 2 3 1 2 --- 1 2 --- 4 12.5 12.5 7–( ) 12.5 8–( ) 12.5 10–( ) 5 12.5 5.5 4.5 2.5××× 773.4375 7WORKEDExample SkillS HEET 4.5 A minute hand moving from the number 12 to the number 4 position sweeps out a sector. If the hand is 10 cm long, what is the area of this sector? THINK WRITE Write down the formula for the area of the sector. Area of sector = Identify the value of the radius. r = 10 cm Calculate the angle of the sector: The angle between the numbers on a clock = 360° ÷ 12 = 30°. From 12 to 4 there are 4 intervals between the numbers. So to ﬁnd the angle of a sector, multiply 30° by 4. θ° = 30° × 4 = 120° Substitute the values of r and θ into the formula and evaluate. Area of sector = × π × 102 = 104.719 755 1 cm2 Write an answer sentence with the number rounded off appropriately and units given. The minute hand as it rotates through an angle of 120° sweeps an area of 104.7 cm2 . 4 12 1 2 3 10 cm1 θ° 360° ----------- πr2× 2 3 4 120 360 --------- 5 8WORKEDExample MQ Maths A Yr 11 - 04 Page 114 Wednesday, July 4, 2001 4:11 PM
15. 15. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 115 Area of composite ﬁgures The term composite means made up of distinct parts. Composite ﬁgures in geometry are ﬁgures comprising a number of distinct shapes. Depending upon the composite ﬁgure, to ﬁnd the overall area you may need to add these individual shapes or subtract one from another. For example, the composite ﬁgure in the diagram at right has been formed using a semicircle and a square. The area of this shape can be found as follows: Area of total ﬁgure = Area of a semicircle (A1) + Area of a square (A2) When ﬁnding the area of a composite ﬁgure, follow the steps given below. 1. Identify the basic shapes that make up the total ﬁgure and number them. 2. Write the expression for the total area in terms of individual shapes. 3. Calculate the area of each individual shape. 4. Add or subtract areas to ﬁnd the total area of the given shape. In the diagram above, we added the areas of the square and the semicircle. If the diagram had been shown as at right, we would have subtracted the area of the semicircle from that of the square. A1 A2 A1 A2 A clock has a minute hand that is 6 cm long and an hour hand that is 3 cm long. In one full revolution of each hand, the minute hand would sweep out a larger circle than the hour hand. What is the difference in the area they cover (to the nearest square centimetre)? THINK WRITE The area required is the area between two circles. Write down the appropriate formula. A = outer area − inner area A = πR2 − πr2 Identify the value of R (radius of larger circle) and the value of r (radius of smaller circle). R = 6, r = 3 Substitute the values of the pronumerals into the formula and evaluate. A = π × 62 − π × 32 = 113.097 − 28.274 = 84.823 cm2 Write an answer sentence with the value rounded to the nearest square centimetre. The difference in area covered by the two hands is approximately 85 cm2 . 3 cm 6 cm R r 1 2 3 4 9WORKEDExample MQ Maths A Yr 11 - 04 Page 115 Wednesday, July 4, 2001 4:11 PM
16. 16. 116 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Calculating areas 1 Copy and complete each of the following. a 70 mm2 = _____ cm2 b 6000 cm2 = _____ m2 c 3 m2 = _____ cm2 d 2.5 km2 = _____ m2 e 4.5 ha = _____ m2 f 3 km2 = _____ ha 2 Find the area of each of the ﬁgures below. a b c d e f g h i remember 1. The area is the space enclosed by the boundaries of a two-dimensional shape. 2. Area is measured in square units. To change from one unit to another, square the appropriate linear measure conversion factor. Multiply by the conversion factor when changing to a smaller unit and divide when converting to a larger unit. 3. Remember these area formulas: Square A = S2 Rectangle A = L × W Parallelogram A = b × h Trapezium A = (a + b)h Triangle A = bh (if perpendicular height is known). If three sides are known, use Heron’s formula A = where s = (a + b + c) Circle A = πr2 Sector A = × πr2 4. For composite ﬁgures, identify shapes of parts of the ﬁgure. After calculating these individual areas, add or subtract them to give the total area. 5. Remember to provide units in the answer. 1 2 --- 1 2 --- s s a–( ) s b–( ) s c–( ) 1 2 --- θ° 360° ----------- remember 4B Work SHEET 4.1 WORKED Example 4 Area of triangle 8 cm 29 mm 3.6 km 3 m 9 m 27mm 38 mm 47 cm 62 cm WORKED Example 5 4.2 m 9.7 m 8.4 km 6.3 km 3.7 m MQ Maths A Yr 11 - 04 Page 116 Wednesday, July 4, 2001 4:11 PM
17. 17. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 117 3 Look at the ﬁgure at right. a Find the area of the outer rectangle. b Find the area of the inner rectangle. c Find the shaded area by subtracting the area of the inner rectangle from the area of the outer rectangle. 4 Find the shaded area in each of the following. a b c d 5 The area of the triangle at right is: j k l m n o p q r s t u A 36 cm2 B 54 cm2 C 108 cm2 D 1620 cm2 WORKED Example 6a 12.8 km 16.9 km 38 mm 87 mm 8 m 80 cm WORKED Example 6b 1 m 12 m 9 m 2.8 m 3.65 m 0.4 m 3.6 cm 9.5 cm 5.4 cm WORKED Example 7 38 cm 27 cm15 cm 12 mm 8 mm7 mm 65 m 16 m58 m WORKED Example 8 45° 6 cm 120° 2 m 300° 5 mm WORKED Example 9 3 m 12 m 20 m 7 m 10 m 8 m14 m 16 m 5 cm 5 cm 9 cm 3 m 9 m 10 m 8 m 5 m 8 m 12 m 12 m 8 m mmultiple choiceultiple choice 15 cm 12 cm 9 cm MQ Maths A Yr 11 - 04 Page 117 Wednesday, July 4, 2001 4:11 PM
18. 18. 118 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 6 Which of the two statements is correct for the two shapes at right? Statement 1. The rectangle and parallelogram have equal areas. Statement 2. The rectangle and parallelogram have equal perimeters. 7 The area of the ﬁgure at right is: 8 Len is having his lounge room carpeted. Carpet costs \$27.80/m2 . The lounge is rectangular with a length of 7.2 m and a width of 4.8 m. a Calculate the area of the lounge room. b Calculate the cost of carpeting the room. 9 A rectangular garden in a park is 15 m long and 12 m wide. A concrete path 1.5 m wide is to be laid around the garden. a Draw a diagram of the garden and the path. b Find the area of the garden. c What are the dimensions of the rectangle formed by the path? d Find the area of concrete needed for the path. 10 Examine the diagram at right. a The circles cover an area of approximately: b The shaded area is approximately: 11 A family-size pizza is cut into 8 equal slices. If the diameter of the pizza is 33 cm, ﬁnd (to the nearest square centimetre) the area of the top part of each slice. 12 The collectable plate shown at right is 22 cm in diameter and has a golden ring that is 0.5 cm wide. Find (to 1 decimal place) the area of the golden ring if its outer edge is 1 cm from the edge of the plate. A Statement 1 B Statement 2 C Both statements D Neither statement A 54 m2 B 165 m2 C 225 m2 D 255 m2 A 402 cm2 B 201 cm2 C 804 cm2 D 805 cm2 E 603 cm2 A 219 cm2 B 421 cm2 C 622 cm2 D 823 cm2 E 220 cm2 mmultiple choiceultiple choice 19 cm 38 cm 38 cm 19 cm mmultiple choiceultiple choice 17 m 7 m 15 m 15 m mmultiple choiceultiple choice 32 cm 22 cm 1 cm 0.5 cm WORKED Example 9 MQ Maths A Yr 11 - 04 Page 118 Wednesday, July 4, 2001 4:11 PM
19. 19. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 119 13 Early-model vehicles had a single wind- screen-wiper blade to remove water from the windscreen. (The bus at left has two single blades of this type.) Using the dimensions given in the diagram: a what area (to the nearest whole number) did the blade cover? b what percentage (to 1 decimal place) of the windscreen was cleared? Calculate the area (or shaded area) of each of the ﬁgures drawn below. Where necessary, give your answer correct to 1 decimal place. 1 2 3 4 5 6 7 8 9 10 140° 45 cm 120 cm 60 cm 1 12 cm 6.3 m 8 cm 30° 62 mm 91 mm 10 cm 4 cm 24 cm 25 cm 30 cm 25 cm 20 cm 40 m 20 m 40 cm 12 cm 76 mm 32 m m 15cm 12 cm 6cm MQ Maths A Yr 11 - 04 Page 119 Wednesday, July 4, 2001 4:11 PM
22. 22. 122 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Patchwork designs Resources: Graph (or unlined) paper, pencil, ruler. In this investigation we look at two common patchwork designs for quilts, and consider the shapes within blocks and the requirements for completing the quilts. Patchwork quilts are frequently constructed from a set of ‘blocks’ of the same pattern. The eight-pointed star block is shown at right. Each block is constructed from a set of smaller shapes sewn together to create the pattern. A border of a complementary colour usually frames the blocks. 1 Identify the shapes in the block above. How many of each shape are required to form the pattern? 2 Each block has dimensions 50 cm square. Sketch a design for a single-bed quilt 173 cm by 218 cm, including the border. How many blocks and how many pieces of each shape would be required? 3 Trace four blocks. Place them together at a point and shade or colour them to illustrate the overall effect of your quilt. 4 Experiment with other patterns incorporating the eight-pointed star block. The quilt displayed provides some suggestions. investigat ioninvestigat ion MQ Maths A Yr 11 - 04 Page 122 Wednesday, July 4, 2001 4:11 PM
23. 23. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 123 Three-dimensional objects As mentioned previously, the shape of 3-D objects can be described by three measure- ments. These objects can be classiﬁed as prisms or non-prisms. Prisms Prisms are 3-D ﬁgures that have a constant cross-section in one direction. This con- stant cross-section is parallel to the face which is called the base of the prism. The name of the prism comes from the shape of its base. Common examples of prisms are: Non-prisms Non-prisms do not have a constant cross-section in any direction. The two types of 3-D object that we will study in this category are pyramids and spheres. Pyramids In pyramids, the cross-section parallel to the base reduces in size as the cross-section progresses from the base to the apex. Common examples of pyramids are: Another interesting block is the cube lattice shown at right. 5 Trace a set of four of these. Place them together at a point and experiment with different shading or colouring of the faces to produce different 3-D effects. Note that dark colours recede and light colours come to the fore in terms of vision. 6 Identify the number and types of different shapes in each block. Draw a design for a single-bed quilt 173 cm by 218 cm (including borders) using blocks of this type. Determine the number of pieces of each shape required. 7 Design a patchwork block of your own. Draw a completed quilt incorporating your block. Cube Rectangular prism Cylinder Square-based pyramid Triangle-based pyramid Cone MQ Maths A Yr 11 - 04 Page 123 Wednesday, July 4, 2001 4:11 PM
24. 24. 124 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Spheres A sphere has no ﬂat faces. When spheres are sliced, the ﬂat surface exposed is always circular. Common examples are: Two properties which three-dimensional ﬁgures possess are surface area and volume. Surface area The surface area of a 3-D object represents the total area of all its exposed faces. To ﬁnd the surface area, we must calculate the area of each face of the object, as identiﬁed by its net, then add all of these areas to ﬁnd the total. The units used for total surface area (TSA) are the same as those used for area. Object Net TSA Cube TSA = 6S2 Rectangular prism TSA = area of TSA = 6 rectangles TSA = 2(WH + LW + LH) Cylinder TSA = area of 2 circles + curved surface = 2πr2 + 2πrH Sphere Hemisphere S 1 3 2 6 S S 4 5 W L H W H H W L L H H r H r r 2πr MQ Maths A Yr 11 - 04 Page 124 Wednesday, July 4, 2001 4:11 PM
25. 25. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 125 Square-based pyramid TSA= area of square + area of 4 triangles = b2 + 4 × ( bh) Cone Curved surface of a cone is formed by removing a sector out of a circle. TSA = area of base (circle) + area of curved surface = πr2 + πrS Sphere Not shown TSA = 4πr2 Hemisphere Open Not shown TSA = 2πr2 Closed Not shown TSA = 2πr2 + area ﬂat circle TSA = 2πr2 + πr2 TSA = 3πr2 Object Net TSA b h b h 1 2 --- S = Slant heightH r Arc length Slant height = radius of sector Circumference of base = arc length of sector r = S Minor sector Major sector Math cad TSA r r MQ Maths A Yr 11 - 04 Page 125 Wednesday, July 4, 2001 4:11 PM
26. 26. 126 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Find the total surface area of the object shown. THINK WRITE Identify the shape. Shape: rectangular prism Write down the formula for the TSA of a rectangular prism. TSA = area 6 rectangles TSA = 2(WH + LW + LH) Allocate a value to the pronumerals. W = 9, H = 17, L = 19 Substitute the values of the pronumerals into the formula. TSA= 2(9 × 17 + 19 × 9 + 19 × 17) Evaluate (brackets ﬁrst, then multiply by 2). = 2(153 + 171 + 323) = 2 × 647 = 1294 Write the answer, including units. TSA= 1294 cm2 17 cm 19 cm 9 cm1 2 3 4 5 6 10WORKEDExample Find the surface area of an open cylindrical can that is 12 cm high and 8 cm in diameter (to 1 decimal place). THINK WRITE Form a net of the open cylinder, transferring all the dimensions to each of the surfaces. (Note that the cylinder has no top surface.) Identify the different-sized common ﬁgures and set up a sum of the surface areas. The length of the rectangle is the circumference of the circle. TSA = A1 + A2 A1 = 2πr × H = 2 × π × 4 × 12 = 301.59 cm2 A2 = π × r2 = π × 42 = 50.27 cm2 Sum the areas. TSA = A1 + A2 = 301.59 + 50.27 = 351.86 cm2 Write your answer. The total surface area of the open cylindrical can is 351.9 cm2 . 8 cm 12cm 1 2 rπ A1 12cm A2 4 cm 2 3 4 11WORKEDExample MQ Maths A Yr 11 - 04 Page 126 Wednesday, July 4, 2001 4:11 PM
27. 27. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 127 Find the surface area of the square pyramid at right. THINK WRITE Calculate the area of the square base. A = S2 = 62 = 36 cm2 Calculate the area of a triangular side. Note each side is identical and the height of each triangular side is 5 cm. A = × b × h = × 6 × 5 = 15 cm2 Calculate the total surface area. (Note: There are 4 identical triangular sides.) SA = 36 + 4 × 15 = 96 cm2 6 cm4cm 5cm 1 2 1 2 --- 1 2 --- 3 12WORKEDExample Find the total surface area of a size 7 basketball with a diameter of 25 cm. Give your answer to the nearest 10 cm2 . THINK WRITE Use the formula for the total surface area of a sphere. Use the diameter to ﬁnd the radius of the basketball and substitute into the formula. Diameter = 25 cm Radius = 12.5 cm TSA of sphere = 4πr2 = 4 × π × 12.52 = 1963.495 Write your answer. Total surface area of the ball is approximately 1960 cm2 . 1 2 13WORKEDExample A die used in a board game has a total surface area of 1350 mm2 . Find the linear dimensions of the die (to the nearest millimetre). THINK WRITE A die is a cube. We can substitute into the total surface area of a cube to determine the dimension of the cube. Divide both sides by 6. TSA = 6 × S2 TSA = 1350 mm2 1350 = 6 × S2 S2 = = 225 Take the square root of both sides to ﬁnd S. S = = 15 mm Write your answer. The dimensions of the die are: 15 mm × 15 mm × 15 mm 1 1350 6 ------------ 2 225 3 14WORKEDExample MQ Maths A Yr 11 - 04 Page 127 Wednesday, July 4, 2001 4:11 PM
28. 28. 128 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d The diagram shows the proposed shape for a new container for takeaway Chinese food. The shape will be used if the TSA of the container is less than 750 cm2 . If the TSA is greater than or equal to 750 cm2 then production and manufacturing costs are too expensive and the takeaway shop will have to stay with the old cylindrical container. Next time I order my Chinese takeaway, could it come in the new design? THINK WRITE Identify the distinct shapes that make up the total object: these are a square-based pyramid and a cube. The base of the pyramid and one face of the cube are not on the surface and therefore their area should not be included. TSA= square pyramid (no base) + 5 faces of a cube Calculate the TSA of a square-based pyramid with no base. (a) Alter the general square-based pyramid formula so as not to include the square base. (b) Allocate a value to the pronumerals. (c) Substitute the values into the formula and evaluate. TSA of square-based pyramid: A = 4 × × b × h b = 10, h = 10 A = 4 × × 10 × 10 = 200 Calculate the TSA of a cube (5 faces only). (a) Alter the general cube formula to include 5 faces instead of 6. (b) Allocate a value to the length. (c) Substitute the value of the side length into the formula and evaluate. TSA of a cube (5 faces only): A = 5S2 S = 10 A = 5 × 102 = 500 Add the individual TSA together to ﬁnd the TSA of the whole object. TSA= 200 + 500 = 700 cm2 Write an answer sentence. Next time I order Chinese takeaway there could be a newly designed container with a surface area of only 700 cm2 . 10 cm 10 cm 1 2 1 2 --- 1 2 --- 3 4 5 15WORKEDExample remember 1. The TSA is the sum of the areas of the outside surfaces of a 3-dimensional object. 2. Formulas for all types of objects are not possible. For those objects without a formula you will need to follow these steps. (i) Draw the net of the object. (ii) Work out the different shapes that make up the net. (iii)Calculate their individual areas. (iv) Add all the individual parts together. 3. Do not include in the TSA the surfaces of contact of the distinct shapes that make up a composite ﬁgure. remember MQ Maths A Yr 11 - 04 Page 128 Wednesday, July 4, 2001 4:11 PM
29. 29. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 129 Total surface area 1 Name each solid in the top row then match it with a net in the bottom row. 2 Draw the net of each of the following solids. 3 Identify the solids from the nets below. Draw the solid for each. 4 Find the total surface area of each of the ﬁgures below. 5 Oliver is making a box in the shape of a rectangular prism. The box is to be 2.5 m long, 1.2 m wide and 0.8 m high. Calculate the surface area of the box. a b c i ii iii a b c a b c a b c d e f 4C 5 cm 9 cm 32 cm WORKED Example 10 3.9 cm 4.1 cm 4 cm 20 cm 13 cm 14 cm 42 mm 7 mm 7 mm MQ Maths A Yr 11 - 04 Page 129 Wednesday, July 4, 2001 4:14 PM
30. 30. 130 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 6 Calculate the surface area of an open box in the shape of a cube, with a side length of 75 cm. (Hint: Since the box is open there are only ﬁve faces.) 7 A room is in the shape of a rectangular prism. The ﬂoor is 5 m long and 3.5 m wide. The room has a ceiling 2.5 m high. The ﬂoor is to be covered with slate tiles, the walls are to be painted blue and the ceiling is to be painted white. a Calculate the area to be tiled. b Each tile is 0.25 m2 . Calculate the number of tiles needed. c Calculate the area to be painted blue. d Calculate the area to be painted white. e One litre of paint covers an area of 12 m2 . How many litres of paint are needed to paint the room? 8 Find the total surface area of the following cylinders. a b c 9 Calculate the surface area of the square pyramid at right. 10 A triangle-based pyramid has four equal faces as shown at right. Calculate the surface area to the nearest cm2 . (Recall Heron’s formula.) WORKED Example 11 Radius = 410 mm Length = 1.5 m 90 cm 28 cm 250 mm 250 mm (Answer to nearest cm2) WORKED Example 12 10 cm 13cm 4 cm MQ Maths A Yr 11 - 04 Page 130 Wednesday, July 4, 2001 4:14 PM
31. 31. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 131 11 Two cubes are drawn such that the side length on the second cube is double the side length on the ﬁrst cube. The surface area of the larger cube will be: A twice the surface area of the smaller cube B four times the surface area of the small cube C six times the surface area of the small cube D eight times the surface area of the small cube. 12 Calculate the surface area of the triangular prism at right. 13 Calculate the surface area of the following cones. 14 Find the total surface area of the following spheres and hemispheres. 15 A cube has a total surface area of 24 cm2 . What is the length of each side? 16 Another cube has a total surface area twice that of the one in question 15. Is the side length of this cube twice that of the one in the previous question? Explain. 17 Calculate the surface area of this prism. 18 A concrete swimming pool is a rectangular prism with the following dimensions: length of 6 metres, width of 4 metres and depth of 1.3 metres. What surface area of tiles is needed to line the inside of the pool? (Give answer in m2 .) Remember there is no top on the pool. a b c a b c mmultiple choiceultiple choice 4 cm 3 cm 5 cm 2 cm 2.4 cm 2.9 cm 40 cm 32 cm 10 cm 18 cm WORKED Example 13 43 mm (Answer in cm2.) 5 m Open (Answer to nearest mm2.) Closed 6.3 mm WORKED Example 14 WORKED Example 15 3.2 m 1m 2 m 4 m 6 m MQ Maths A Yr 11 - 04 Page 131 Friday, July 6, 2001 8:40 AM
32. 32. 132 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 19 What is the total area of canvas needed for the tent (including the base) shown in the diagram at right? Give the answer to the nearest m2 . 20 A ball used in a game of pool has a diameter of 48 mm. The total surface area of the ball is closest to: A 1810 mm2 B 2300 mm2 C 7240 mm2 D 28 950 mm2 E 115 800 mm2 21 The total surface of a golf ball of radius 21 mm is closest to: 22 The formula for the total surface area for the object shown is: A abh B 2 × bh + ab + 2 × ah C 3( bh + ab) D bh + 3ab E bh + 3ab 23 The total surface area of a poster tube that is 115 cm long and 8 cm in diameter is closest to: 24 A baker is investigating the best shape for a loaf of bread. The shape with the smallest surface area stays freshest. The baker has come up with two shapes: a rectangular prism with a 12-cm-square base and a cylinder with ﬂat, round ends that have a 14-cm diameter. a Which shape stays fresher if they have the same overall length of 32 cm? b What is the difference between the total surface areas of the two loaves of bread? A 550 mm2 B 55 cm2 C 55 000 mm2 D 0.055 m2 E 5.5 cm2 A 3000 cm2 B 2900 cm2 C 1500 mm2 D 6200 m2 E 23 000 cm2 4.5 m 6.5 m 2.5 m 1.5m 1.0m mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice h b a 1 2 --- 1 2 --- 1 2 --- 1 2 --- mmultiple choiceultiple choice MQ Maths A Yr 11 - 04 Page 132 Wednesday, July 4, 2001 4:14 PM
33. 33. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 133 1 Calculate the area of a rectangle with a length of 0.4 m and a width of 1.1 m. 2 Calculate the area of a triangle with a base of 12.3 m and a height of 4.8 m. 3 Calculate the area of the trapezium at right. Name the solids below. 4 5 6 7 Find the surface area of the cube shown at left. 8 Find the surface area of a rectangular prism with a length of 8 cm, a width of 5 cm and a height of 6 cm. 9 Find the surface area of the triangular prism below. 10 Find the surface area of the square pyramid below. 2 32 m 96 m 56 m 10 cm 8 cm 20 cm 6 cm 6cm 8 cm 9 cm MQ Maths A Yr 11 - 04 Page 133 Wednesday, July 4, 2001 4:14 PM
34. 34. 134 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Volume and capacity Volume The volume of a 3-D object represents the amount of space contained in, or occupied by, the object. The units used to measure volume are those of cubic measure: mm3 , cm3 , m3 . Constructing our conversion ladder, as before, will enable us to convert from one unit to another. To obtain the conversion factors for cubic measure, the linear measure conversion factors are cubed. The same procedure applies as before: multiply when converting to a smaller unit and divide when converting to a larger unit. The capacity of a 3-D object refers to the quantity of solid, liquid or gas it could hold. The units used to measure capacity are millilitres (mL), litres (L) and kilolitres (kL). The conversion ladder for capacity units is as follows: In capacity units, 1 mL represents the amount of liquid which a 1-cm cube could hold. Two useful conversion relationships are: 1 cm3 ≡ 1 mL and, for larger quantities 1 m3 ≡ 1 kL mm3 10 × 10 × 10 = 1000 cm3 100 × 100 × 100 = 1000 000 m3 × ÷ mL 1000 L 1000 kL × ÷ EXCE L Spreadshe et Capacity Mat hcad Unit con- versions Convert 1.12 cm3 to mm3 . THINK WRITE The conversion from centimetres to millimetres is 1 cm = 10 mm. The conversion factor for cm3 to mm3 is to multiply by 103 or 1000; that is, 1 cm3 = 1000 mm3 . 1.12 cm3 = 1.12 × 1000 mm3 = 1120 mm3 Write the answer in correct units. 1 2 3 16WORKEDExample 1 cm 1 cm 1 cm Holds 1 mL MQ Maths A Yr 11 - 04 Page 134 Wednesday, July 4, 2001 4:14 PM
35. 35. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 135 In our calculations of volume, we shall consider only three classes of 3-D ﬁgures: prisms, pyramids and spheres. Volume of prisms The volume of a prism is given by the following formula: Volume of a prism = cross-sectional area × height of the prism The height is the dimension perpendicular to the cross-sectional area. Shape Cross-sectional shape Volume Cylinder Area = πr2 V = area of a circle × height = πr2 × H Triangular prism Area = bh V = area of a triangle × height = bh × H Note: Lower-case h represents the height of the triangle. Rectangular prism Area = L × W V = area of a rectangle × height = L × W × H Cube Area = S2 V = area of a square × height = S2 × H = S2 × S = S3 (since in a cube, H = S) Math cad Capacity Convert: a 400 cm3 into mL b 1200 cm3 into mL and into L c 2 kL into m3 . THINK WRITE a Since 1 cm3 is equivalent to 1 mL, then 400 cm3 is equivalent to 400 mL. a 400 cm3 = 400 mL b Each 1 cm3 will hold 1 mL of liquid. Therefore, 1200 cm3 will hold 1200 mL of liquid. b 1200 cm3 = 1200 mL To change mL to L, divide by 1000 (since there are 1000 mL in 1 L). = 1.2 L c One kL is equivalent to 1 m3 . Therefore, 2 kL is equivalent to 2 m3 . c 2 kL = 2 m3 1 2 17WORKEDExample H r r b Hh b h 1 2 --- 1 2 --- E XCEL Spread sheet VolumeWL H W L Math cad Volume formulasS H S MQ Maths A Yr 11 - 04 Page 135 Wednesday, July 4, 2001 4:14 PM
36. 36. 136 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Volume of pyramids As we have seen previously, a pyramid has a ﬂat base at one end, and tapers to a point at the other. Some examples of pyramids are shown below. A cone is really a circle- based pyramid. Cone Square pyramid Rectangular pyramid Triangular pyramid Comparing volumes of pyramids and prisms Resources: Set of 3-D volumetric shapes of pyramids and prisms with same base area and height; water (or rice). For the following investigation, the volumes of pairs of open 3-D containers are compared by considering the amount of water (or rice) each can hold. Each 3-D pair should have the same base area and the same perpendicular height. Consider the following pairs of containers: square-based pyramid and cube rectangle-based pyramid and rectangular prism triangle-based pyramid and triangular prism cone and cylinder 1 Fill the ﬁrst container with the water (or rice), then pour the contents into the second container. Continue reﬁlling the ﬁrst container and pouring the contents into the second until the second container is full. How many times was it necessary to do this? Find the volume of the shape shown correct to 1 decimal place. THINK WRITE Identify the shape. Triangular prism Write down the appropriate formula for the volume. V = bh × H Allocate values to the pronumerals keeping in mind that b and h are the base and height of a triangular cross-section or base of the prism, while H is the height of the prism. b = 2.6, h = 2.3, H = 3.2 Substitute and evaluate, rounding the answer to 1 decimal place. V = × 2.6 × 2.3 × 3.2 = 9.568 ≈ 9.6 m3 2.6 m 3.2 m 2.3m 1 2 1 2 --- 3 4 1 2 --- 18WORKEDExampleinv estigat ioninv estigat ion MQ Maths A Yr 11 - 04 Page 136 Wednesday, July 4, 2001 4:14 PM
37. 37. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 137 A pyramid does not have a uniform cross-section. The cross-sectional area becomes smaller as it nears the apex (point). The internal capacity or volume of a tapered object is a fraction of the volume of a prism. Mathematicians found this fraction to be a third ( ). They deﬁned the base of a pyramid to be the ﬂat end opposite the apex. To calcu- late the volume of a pyramid we ﬁnd the area of the ﬂat end, multiply this by the height of the pyramid (which must be perpendicular to the base) and then multiply by (or divide by 3). Volume of a pyramid = × area of base × height of object The following table shows the formulas for the volume of some common pyramids. 2 From your results, how would you compare the volume of a pyramid with that of a prism of the same base area and height? 3 Having previously considered the general formula for the volume of a prism, suggest a general formula for the volume of a pyramid. 4 Draw labelled diagrams and deduce speciﬁc formulas for the volume of each of the following: a square-based pyramid b rectangle-based pyramid c triangle-based pyramid d cone (circle-based pyramid). Shape Flat end (base) shape Volume Cone V = × area of a circle × height V = πr2 × H Square pyramid V = × area of a square × height V = S2 × H Rectangular pyramid V = × area of a rectangle × height = L × W × H Triangular pyramid V = × area of a triangle × height V = ( bh) × H Note: Lower-case h represents the height of the triangle. 1 3 --- 1 3 --- 1 3 --- Math cad Mensura- tion r H r 1 3 --- 1 3 --- H S S 1 3 --- 1 3 --- H L W W L 1 3 --- 1 3 --- Math cad Volume formulasH h b h b 1 3 --- 1 3 --- 1 2 --- MQ Maths A Yr 11 - 04 Page 137 Wednesday, July 4, 2001 4:14 PM
38. 38. 138 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Volume of spheres The volume of a sphere of radius r is given by the following formula: Volume of a sphere = πr3 A hemisphere is half of a sphere. Its volume, therefore, is half of the volume of a sphere. Volume of hemisphere = (volume of sphere) = × πr3 = πr3 Find the volume of the pyramid at right (to the nearest m3 ). THINK WRITE Write the equation. V = × area of base × height The pyramid has a square base. It is a square pyramid. The area of the base is given by S2 . Area of base = S2 Calculate the volume. Volume = × S2 × H Volume = × 302 × 40 Volume = 12 000 m3 Write your answer. The volume of the square pyramid is 12 000 m3 . 30 m 30 m Height of pyramid = 40 m 1 1 3 --- 2 3 1 3 --- 1 3 --- 4 19WORKEDExample Find the volume of the cone at right, correct to 2 decimal places. THINK WRITE Write the formula. V = πr2 H Substitute the radius and height. = × π × 3.22 × 8.5 Calculate the volume. = 91.15 cm3 3.2 cm 8.5 cm 1 1 3 --- 2 1 3 --- 3 20WORKEDExample Mat hcad Volume formulas r 4 3 --- 1 2 --- r 1 2 --- 4 3 --- 2 3 --- MQ Maths A Yr 11 - 04 Page 138 Wednesday, July 4, 2001 4:14 PM
39. 39. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 139 Note that the volume and capacity of a 3-D object do not depend on whether the object is open or closed. An open rainwater tank could hold the same quantity of water as a closed one. The surface area, however, varies depending on whether the object is open or closed. Cross-sections of solid 3-D shapes Resources: Plasticine (or play dough), thread. In this activity we investigate the surfaces exposed when solid 3-D shapes are cut in different directions. 1 Using plasticine, mould 3-D solids in the shape of a: a cube b rectangular prism c triangular prism d cylinder e sphere. 2 Investigate slicing these solids in various directions with a piece of thread to see whether it is possible to obtain sections with faces in the shape of a: a square b rectangle c triangle d ellipse e circle. 3 Copy and complete the following table, showing the direction of the sectional cut required for each particular face to be exposed. Not all face shapes are possible. Find the volume of a sphere with a radius of 9.5 cm, correct to the nearest cm3 . THINK WRITE Write the formula. V = πr3 Substitute the radius. = × π × 9.53 Calculate the volume. = 3591 cm3 1 4 3 --- 2 4 3 --- 3 21WORKEDExample inv estigat ioninv estigat ion Section face Solid Cube Rectangular prism Triangular prism Cylinder Sphere Square Rectangle Triangle Ellipse Circle MQ Maths A Yr 11 - 04 Page 139 Wednesday, July 4, 2001 4:14 PM
40. 40. 140 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Volume of composite objects Often, an object can be identiﬁed as comprising two or more different common prisms, pyramids or spheres. Such ﬁgures are called composite objects. The volume of a com- posite object is found by adding the volumes of the individual common ﬁgures or deducting volumes. A grain silo can be modelled as the sum of a cylinder and a large cone, less the tip of the large cone. The volume of a composite object is equal to the sum (or difference) of the individual common prisms, pyramids or spheres. Vcomposite = V1 + V2 + V3 + . . . (or Vcomposite = V1 − V2) Find the capacity of the object shown at right (to the nearest litre). THINK WRITE The object is a composite of a cylinder and a square prism. The volume of the composite object is the sum of volumes of the cylinder plus the prism. Vcomposite = volume of cylinder + volume of square prism = (πr2 × Hc) + (S2 × Hs) = (π × 62 × 20) + (182 × 25) = 2261.946 711 + 8100 = 10 361.946 711 cm3 Convert to litres using the conversion of 1 cm3 = 1 mL 1000 mL = 1 L 10 362 cm3 = 10.362 litres Write your answer. The capacity of the object is 10 litres. 25cm 12 cm 18 cm 20cm 1 r = 6 cm 18 cm 25cm 18 cm H=20cm 2 3 22WORKEDExample MQ Maths A Yr 11 - 04 Page 140 Wednesday, July 4, 2001 4:14 PM
41. 41. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 141 Volume and capacity 1 Convert the volumes to the units speciﬁed. 2 Convert the following units as indicated. 3 Calculate the volume and capacity of each of the prisms below. a 0.35 cm3 to mm3 b 4800 cm3 to m3 c 56 000 cm3 to litres d 1.5 litres to cm3 e 1.6 m3 to litres f 0.0023 cm3 to mm3 g 0.000 57 m3 to cm3 h 140 000 mm3 to litres i 250 000 mm3 to cm3 a 750 cm3 = mL b 800 cm3 = L c 2500 cm3 = mL d 40 000 cm3 = L e 6 m3 = cm3 = mL = L f 12 m3 = L g 4.2 m3 = kL h 7.5 m3 = kL = L i 5.2 mL = cm3 j 6 L = cm3 k 20 L = mL = cm3 l 5.3 kL = m3 a b c remember 1. Volume is measured in cubic units: mm3 , cm3 , m3 . 2. Capacity is measured in mL, L, kL. 3. To convert volume units into capacity units: 1 cm3 = 1 mL 1 m3 = 1 kL 4. The volume of a prism is found using the formula V = area of base × height of prism 5. The volume of a pyramid is found using the formula V = area of base × height of pyramid 6. The volume of a sphere is calculated using V = πr3 and for a hemisphere, V = πr3 7. The volume of a container is not dependent on whether it is open or closed. 8. The volume of a composite object can be found by adding or subtracting volumes of individual prisms, pyramids or spheres. 1 3 --- 4 3 --- 2 3 --- remember 4D Work SHEET 4.2WORKED Example 16 WORKED Example 17 5 cm 2.4 m 13 m MQ Maths A Yr 11 - 04 Page 141 Wednesday, July 4, 2001 4:14 PM
42. 42. 142 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 4 For each of the triangular prisms below ﬁnd: i the area of the base of the prism ii the volume of the prism. a b c d 5 The shape at right could be described as a: 6 The area of the base of a prism is 34.67 cm2 , and the height is 3.6 cm. The volume of the prism is: 7 The dimensions of a rectangular prism are all doubled. The volume of the prism will increase by a factor of: 8 A refrigerator is in the shape of a rectangular prism. The internal dimensions of the prism are 60 cm by 60 cm by 140 cm. a Find the volume of the refrigerator in cm3 . b The capacity of a refrigerator is measured in litres. If 1 cm3 = 1 mL, ﬁnd the capacity of the refrigerator in litres. d e f g h i A cube B square prism C rectangular prism D both B and C A 38.27 cm2 B 38.27 cm3 C 124.12 cm2 D 124.812 cm3 A 2 B 4 C 6 D 8 4.2 m 3.2 m 50 mm 9 mm 9 mm 12.5 m 20.5 m 16.5 m 6 cm 12 cm 12 m 3 m 13 cm 27 cm WORKED Example 18 8 cm 3 cm 5 cm 6 cm 8 cm 12 cm 3.4 m 2.7 m 1.5 m 3.2 m 12.5 m 7.8 m mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 04 Page 142 Wednesday, July 4, 2001 4:14 PM
43. 43. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 143 9 A semitrailer is 15 m long, 2.5 m wide and 2.7 m high in the shape of a rectangular prism. Find the capacity of the semitrailer. (Ignore the thickness of the walls.) 10 A petrol tanker is shown at right. The tank is cylindrical in shape. The radius of the tank is 2 m and the length is 12 m. Ignoring the thickness of the material, calculate: a the volume of the tank, correct to 3 decimal places b the capacity of the tank, to the nearest 100 litres. (1 m3 = 1000 L). 11 At right is a diagram of a concrete slab for a house. a Calculate the area of the slab. b The slab is to be 10 cm thick. Calculate the volume of concrete needed for the slab. (Hint: Write 10 cm as 0.1 m.) c Concrete costs \$145.50/m3 to lay. Calculate the cost of this slab. 12 A ﬂat rectangular roof is 14 m long and 8 m wide. When it rains, the water is col- lected in a cylindrical tank. a Calculate the volume of water collected on the roof when 25 mm of rain falls. b How many litres of water does the roof collect? c The cylindrical tank has a radius of 1.8 m and is 2.4 m high. What is the capacity of the tank, in litres? d By how much does the depth of water in the tank rise when the rain falls? Answer in centimetres, correct to 1 decimal place. 13 For each of the following pyramids, calculate the volume by ﬁrst calculating the area of the base shape. a b c d 12 m 2 m 10 m 10 m 2.5 m 15 m WORKED Example 19 8 cm 6 cm 15 cm 8 cm 14 cm 10 m 12 m 6 m 12 cm 5 cm 8 cm 6 cm MQ Maths A Yr 11 - 04 Page 143 Wednesday, July 4, 2001 4:14 PM
44. 44. 144 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 14 Find the volume of each of the following cones, correct to the nearest whole number. a b c d 15 A cone has a base with a diameter of 9 cm and a height of 12 cm. Calculate the volume of that cone, correct to 1 decimal place. 16 Calculate the volume of each of the following spheres, correct to 1 decimal place. a b c d 17 Calculate the volume of a sphere with a diameter of 2.3 cm. Answer correct to 2 decimal places. 18 Which of the following solids could not be described as a pyramid? A B C D 19 A triangular pyramid and a square pyramid both have a base area of 20 cm2 and a height of 15 cm. Which has the greater volume? A the triangular pyramid B the square pyramid C both have equal volume D this can’t be calculated WORKED Example 20 5 cm 10 cm 12 cm 12 cm 8 mm 33 mm 42 cm 42 cm WORKED Example 21 6 cm 8 cm 12.5 m 3.2 m mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 04 Page 144 Wednesday, July 4, 2001 4:14 PM
45. 45. C h a p t e r 4 L e n g t h , a r e a a n d v o l u m e 145 20 A spherical balloon has a volume of 500 cm3 . It is then inﬂated so that the diameter of the balloon is doubled. The volume of the balloon will now be: 21 In each of the following, the prism’s front face is made up of a composite ﬁgure. For each: i calculate the area of the front face ii ﬁnd the volume of the prism. a b c d 22 Find the volume of the solid at right. Answer correct to 1 decimal place. 23 A hollow rubber ball is to be made with a radius of 8 cm, and the rubber to be used is 1 cm thick. a What would be the radius of the hollow inside? b Calculate the volume of the ball. c Calculate the volume of space inside the ball. d Calculate the amount of rubber (in cm3 ) needed to make the ball. 24 The ﬁgure at right is a truncated cone, that is, a cone with the top cut off. a Calculate the volume of the cone before it was truncated. b The portion cut off was itself a cone. Calculate its volume. c Calculate the volume of the truncated cone. 25 Use the same method as in question 24 to ﬁnd the volume of the truncated pyramid shown at right. 26 The ﬁgure at right is of an ice-cream cone, containing a spherical scoop of ice-cream (a whole sphere). a Calculate the volume of the cone. b Calculate the volume of the scoop of ice-cream. c Calculate the total volume of the shape. (Hint: Only half the sphere sits above the cone.) A 1000 cm3 B 2000 cm3 C 3000 cm3 D 4000 cm3 mmultiple choiceultiple choice 10 cm 10 cm20 cm 4 cm 16 cm 8 m 9 m 4 m 6 cm 20 cm 15 cm 8 cm 12 cm 3 m 18 m 12 m 12 m 6 m WORKED Example 22 4 cm 12 cm 6 cm 3 cm 15 cm 6 cm 1 cm 5 cm 3 cm 3 cm 8 cm 2.5 cm MQ Maths A Yr 11 - 04 Page 145 Wednesday, July 4, 2001 4:14 PM