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Volume Practice Menu © Boardworks Ltd 2001 Volumes This unit explains how the volumes of various solids are calculated. It includes simple applications of formula and clear examples. It also contains a variety of challenges and problems. Menu © Boardworks Ltd 2001 Volumes What 3-D solid is this? Menu © Boardworks Ltd 2001 Volumes What 3-D solid is this? Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Menu Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Formulae Summary Cuboid = width x length x height Prism = area of end x height Cylinder = r2 x height Cone 1 = 3 Pyramid 1 = 3 base area x height r2 x height Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 PACKETS IN A BOX PROBLEMS In each problem, find the number of smaller packets that will fit neatly into the box. 1 8 7 6 5 4 3 2 Menu © Boardworks Ltd 2001 PACKETS IN A BOX EXAMPLE 1 In this problem, find the number of smaller packets that will fit neatly into the box. Packet 2cm 4cm Box Dimensions 2cm 12 x 4 x 6 4 x 2 x 2 3 x 2 x 3 = 18 12cm x 4cm x 6cm Answer 18 Menu © Boardworks Ltd 2001 PACKETS IN A BOX EXAMPLE 2 In this problem, find the number of smaller packets that will fit neatly into the box. Packet Box Dimensions 3cm 5cm 2cm 20 x 8 x 6 5 x 2 x 3 4 x 4 x 2 = 32 20cm x 8cm x 6cm Answer 32 Menu © Boardworks Ltd 2001 PACKETS IN A BOX EXAMPLE 3 Be careful with this one! There is an extra step! Packet 2cm 3cm Box Dimensions 9cm x 8cm x 25cm 5cm 9 x 8 x 25 3 x 2 x 5 3 x 4 x 5 = 60 Answer Note change of order !! 60 Menu © Boardworks Ltd 2001 PACKETS IN A BOX PROBLEMS 1 In each problem, find the number of smaller packets that will fit neatly into the box. 1 Box 10 x 8 x 6 Packet 5 x 2 x 3 2 x 4 x 2 = 16 2 Box 12 x 15 x 8 Packet 2 x 3 x 2 6 x 5 x 4 = 120 3 Box 30 x 8 x 10 Packet 6 x 4 x 5 5 x 2 x 2 = 20 4 Box 50 x 20 x 15 Packet 5 x 4 x 3 10 x 5 x 5 = 250 Menu © Boardworks Ltd 2001 PACKETS IN A BOX PROBLEMS 2 In each problem, find the number of smaller packets that will fit neatly into the box. 1 Box 9 x 10 x 10 Packet 3 x 2 x 5 3 x 5 x 2 = 30 2 Box 24 x 25 x 30 Packet 6 x 5 x 3 4 x 5 x 10 = 200 3 Box 40 x 20 x 20 Packet 5 x 4 x 2 8 x 5 x 10 = 400 4 Box 18 x 21 x 15 Packet 3 x 3 x 3 6 x 7 x 5 = 210 Menu © Boardworks Ltd 2001 PACKETS IN A BOX PROBLEMS 3 In some of these problems you may have to re-order the numbers! 1 Box 50 x 24 x 16 Packet 5 x 4 x 4 5 x 4 x 4 = 80 2 Box 15 x 6 x 8 Packet 52 x 5 2 x 2 3 x 3 x 4 = 36 3 Box 20 x 18 x 14 Packet 65 x 7 6 xx 75 4 x 3 x 2 = 24 4 Box 28 x 40 x 15 Packet 75 x 7 4 x 4 5 7 x 10 x 3 = 210 Menu © Boardworks Ltd 2001 PACKETS IN A BOX PROBLEMS 4 In each problem, find the number of smaller packets that will fit neatly into the box. 1 Box 24 x 20 x 90 5 x 9 Packet 8 5 x 89 3 x 4 x 10 = 120 2 Box 30 x 80 x 25 5 x 3 Packet 3 8 x 85 10 x 10 x 5 = 500 3 Box 75 x 100 x 60 Packet 25 6 x 25 20 x 20 6 3 x 5 x 10 = 150 4 Box 70 x 40 x 30 Packet 15 7 x 7 8 x 15 8 10 x 5 x 2 = 100 Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Menu Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 GOING FOR GOLD PROBLEM This problem requires knowledge of how to calculate the volume of a “trapezoidal prism” and a “cylinder”. Look up the relevant sections if some background work is necessary. Menu © Boardworks Ltd 2001 GOING FOR GOLD PROBLEM In a bullion robbery, a gang of thieves seize 100 gold ingots with the dimensions shown on the right. Find the volume of one ingot. 6cm 5cm 25cm 10cm The 100 ingots are melted down and made into 3cm 0.5cm souvenir medals. How many medals could be produced? Menu © Boardworks Ltd 2001 GOING FOR GOLD PROBLEM STEP 1 Find volume of one ingot. 6cm Area of End 1 = (10 + 6) x 5 = 40 cm2 2 Vol = 40 x 25 = 1000 cm3 STEP 2 Volume of 100 ingots = 100 x 1000 =100000 cm3 5cm 25cm 10cm Volume of Prism = Area of End x Length Menu © Boardworks Ltd 2001 GOING FOR GOLD PROBLEM STEP 3 Find volume of one disc using the formula for a “cylinder”. Vol = x 3 x 3 x 0.5 = 14.13 cm3 STEP 4 Vol. of gold = 100000 cm3 Number of “medals” ? = 100000 14.13 3cm 0.5cm Volume of a Cylinder ? = r2 x height = 7077 (approx) Menu © Boardworks Ltd 2001 GOING FOR GOLD PROBLEM 2 Similar problem ... Different numbers!!! In a bullion robbery, a gang of thieves seize 500 gold ingots with the dimensions shown on the right. Find the volume of one ingot. 4cm 3cm 10cm 6cm The 500 ingots are melted down and made into 2cm 0.5cm souvenir medals. How many medals could be produced? Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Menu Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Half Full Problem There is a famous saying concerning the way different people look at situations. “The glass is half full v the glass is half empty” BUT ... What do we mean by HALF FULL??? Menu © Boardworks Ltd 2001 Half Full Problem 4cm 8cm See next slide for hints. 1 This ice cream cone has a height of 8cm and circular face of radius 4cm. When full it contains 134 cm3. (Check) What height will the ice cream be at when it is half full? Menu © Boardworks Ltd 2001 Half Full Problem Half the height. Half the circle radius. 2 cm 4cm Try other measurements 2 Half the volume = half of 134 cm3 = 67 cm3 BUT what do you notice when you find the volume of this “half sized” cone? Only ... 16.6 cm3 Menu © Boardworks Ltd 2001 Half Full Problem The circle radius is half the height. 3cm 6cm 3 TARGET VOLUME = half of 134cm3 67 cm3 This one on the left gives a volume of ... 56.5 cm3 Is this any closer? Try others!! Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Cuboid Example 1 Volume of a cuboid = width x length x height Vol of cuboid 4 cm = 5 x 3 x 4 Vol = 60 cm3 3 cm 5 cm Menu © Boardworks Ltd 2001 Cuboid Example 2 Volume of a cuboid = width x length x height Vol of cuboid = 10 x 3 x 4 Vol = 120 cm3 4 cm 10 cm 3 cm Menu © Boardworks Ltd 2001 Cuboid Example 3 Volume of a cuboid = width x length x height Vol of cuboid 6 cm = 6 x 6 x 6 Vol = 216 cm3 6 cm 6 cm Menu © Boardworks Ltd 2001 Cuboid Example 4 Volume of a cuboid = width x length x height Vol of cuboid 4 cm = 8 x 5 x 4 Vol = 160 5 cm cm3 8 cm Menu © Boardworks Ltd 2001 Cuboids Basic Exercise A 1 2 36cm3 6cm 3cm 32cm3 2cm 3 8cm 2cm 2cm 4 36cm3 4cm 3cm 3cm 56cm3 7cm 4 cm 2 cm Menu © Boardworks Ltd 2001 Cuboids Basic Exercise B 1 2 60cm3 4cm 48cm3 3cm 8cm 5cm 3 3cm 2cm 4 120cm3 6cm 5cm 4cm 21cm3 3.5cm 3 cm 2 cm Menu © Boardworks Ltd 2001 Cuboids Gaps Exercise A Fill in the missing values. Width Length Height Volume 1 2 3 5 30 cm3 2 4 5 10 200 cm3 3 3 3 5 45 cm3 4 5 2 6 60 cm3 5 4 8 10 320 cm3 Menu © Boardworks Ltd 2001 Cuboids Gaps Exercise B Fill in the missing values. Width Length Height Volume 1 4 5 8 160 cm3 2 8 2 3 48 cm3 3 5 10 20 1000 cm3 4 4 6 10 240 cm3 5 6 5 9 270 cm3 Menu © Boardworks Ltd 2001 Cuboids Gaps Exercise C Fill in the missing values. Width Length Height Volume 1 2.5 5 10 125 cm3 2 4 6 4 96 cm3 3 6 6 6 216 cm3 4 5 4 8 160 cm3 5 6 8 20 960 cm3 Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Triangular Prism Example 1 Volume of a prism = area of end x length Area of End 1 = (5 x 4) = 10 cm2 2 4cm Vol = 10 x 8 = 80 cm3 8cm 5cm Menu © Boardworks Ltd 2001 Triangular Prism Example 2 Volume of a prism = area of end x length Area of End 1 = (2 x 6) = 6 cm2 2 Vol = 6 x 7 = 42 cm3 2cm 7cm 6cm Menu © Boardworks Ltd 2001 Triangular Prism Example 3 Volume of a prism = area of end x length Area of End = 1 (4 x 3) = 6 cm2 2 Vol = 6 x 9 = 54 cm3 3cm 9cm 4cm Menu © Boardworks Ltd 2001 Triangular Prism Example 4 Volume of a prism = area of end x length Area of End = 1 (5 x 6) =15 cm2 2 Vol = 15 x 12 = 180cm3 12cm 6cm 5cm Menu © Boardworks Ltd 2001 Triangular Prisms 1 6cm3 4cm 1.5cm Basic Ex A 2 4cm 45cm3 3cm 6cm 4cm 4cm 2cm 3 32cm3 5cm 4 24cm3 2cm 6cm 4cm Menu © Boardworks Ltd 2001 Triangular Prisms 1 35cm3 7cm 2cm Basic Ex B 2 30cm3 5cm 4cm 3cm 5cm 3 105cm3 5cm 6cm 7cm 4 105cm3 3cm 10cm 7cm Menu © Boardworks Ltd 2001 Right Angle Triangular Prisms Ex A Fill in the missing values. Base Height Length Volume 1 2 5 6 30 cm3 2 2 4 4 16 cm3 3 2 5 10 50 cm3 4 3 4 5 30 cm3 5 5 5 8 100 cm3 Menu © Boardworks Ltd 2001 Right Angle Triangular Prisms Ex B Fill in the missing values. Base Height Length Volume 1 4 5 6 60 cm3 2 2 6 3 18 cm3 3 6 10 10 300 cm3 4 5 5 10 125 cm3 5 3 5 10 75 cm3 Menu © Boardworks Ltd 2001 Right Angle Triangular Prisms Ex C Fill in the missing values. Base Height Length Volume 1 3 5 8 60 cm3 2 3 3 5 22.5 cm3 3 8 6 20 480 cm3 4 4 6 10 120 cm3 5 6 5 9 135 cm3 Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Trapezoidal Prism Example 1 Volume of a prism = area of end x length Area of End = 1 (4 + 2) x 3 = 9 cm2 2 Vol = 9 x 5 = 45 cm3 2cm 3cm 5cm 4cm Menu © Boardworks Ltd 2001 Trapezoidal Prism Example 2 Volume of a prism = area of end x length Area of End = 1 (8 + 4) x 6 = 36 cm2 2 4cm 6cm 10cm Vol = 36 x 10 = 360 cm3 8cm Menu © Boardworks Ltd 2001 Trapezoidal Prism Example 3 Volume of a prism = area of end x length Area of End = 1 (3 + 2) x 5 =12.5 cm2 2 Vol = 12.5 x 8 = 100 cm3 2cm 8cm 5cm 4cm 3cm Menu © Boardworks Ltd 2001 Trapezoidal Prisms 1 48cm3 Basic Ex A 2 2cm 3cm 2cm 6cm 4cm 5cm 4cm 5cm 3 60cm3 160cm3 4cm 4cm 6cm 8cm 4 10cm3 4cm 2cm 1cm 3cm Menu © Boardworks Ltd 2001 Trapezoidal Prisms 1 180cm3 Basic Ex B 2 3cm 4cm 3cm 10cm 10cm 5cm 5cm 8cm 3 200cm3 144cm3 4cm 3cm 6cm 8cm 4 10cm 55cm3 4cm 1cm 7cm Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Cylinders Cylinders occur a lot in everyday life. Many containers are this shape. Volume of a cylinder ??? = r2 x height Menu © Boardworks Ltd 2001 Cylinder Example 1 Volume of a cylinder = r2 x height Volume of cylinder = x3x3x6 Vol = 169.65 cm3 6cm 3cm Menu © Boardworks Ltd 2001 Cylinder Example 2 Volume of a cylinder = r2 x height Volume of cylinder = x2x2x8 Vol = 100.53 cm3 8cm 2cm Menu © Boardworks Ltd 2001 Cylinder Example 3 Volume of a cylinder = r2 x height Volume of cylinder = x5x5x4 4cm 5cm Vol = 314.16 cm3 Menu © Boardworks Ltd 2001 Cylinders 1 Basic Exercise A 226.2cm3 2 307.9cm3 7cm 3cm 3 2cm 3c 8cm 197.9cm3 4 8cm 37.7cm3 7cm 3cm 1cm 12cm Menu © Boardworks Ltd 2001 Cylinders 1 Basic Exercise B 88.0cm3 2 3cm 141.4cm3 5cm 2cm 3 7cm 3c 42.4cm3 6cm 1.5cm 4 8cm 235.6cm3 5cm 3cm Menu © Boardworks Ltd 2001 Volumes - Contents List of Formulae A. Problems / Challenges Problems and challenges involving the formulae for the volumes of a variety of shapes. Packets in a Box Challenge (Cuboids) Ingots Problem (Prisms and Cylinders) Half Full Problem (Cones) Each of the below sections is a a mixture of explanations and basic consolidation activities B. Cuboids C. Triangular Prisms D. Trapezoidal Prisms E. Cylinders F. Cones and Pyramids Menu © Boardworks Ltd 2001 Cone Example 1 Volume of a cone 1 = r 2 x height 3 Volume of cone 1 = 3 x3x3x6 Vol = 56.55 cm3 6cm 3cm Menu © Boardworks Ltd 2001 Cone Example 2 Volume of a cone 1 = r 2 x height 3 Volume of cone 1 = 3 x4x4x5 Vol = 83.78 cm3 5cm 4cm Menu © Boardworks Ltd 2001 Cone Example 3 Volume of a cone 1 = r 2 x height 3 Volume of cone 1 = 3 x2x2x8 8cm Vol = 33.51 cm3 2cm Menu © Boardworks Ltd 2001 Pyramid Example 1 Volume of a pyramid 1 = x base area x height 3 Vol of pyramid 1 = x 6 x 6 x 10 3 10cm Vol = 120cm3 6cm Menu © Boardworks Ltd 2001 Pyramid Example 2 Volume of a pyramid 1 = x base area x height 3 Vol of pyramid 1 = x8x8x6 3 Vol = 128 cm3 6cm 8cm Menu © Boardworks Ltd 2001 Pyramid Example 3 Volume of a pyramid 1 = x base area x height 3 Vol of pyramid 1 = x 10 x 10 x 4 3 Vol = 133.33 cm3 4cm 10cm Menu © Boardworks Ltd 2001 Cones and Pyramids 1 47.1cm3 Basic Ex A 2 9 cm3 3cm 5cm 3cm 3cm 3 150.8cm3 4 8cm 3c 130.7cm3 8cm 4cm 6cm 7cm Menu © Boardworks Ltd 2001 Cones and Pyramids 1 402.1cm3 6cm Basic Ex B 2 64 cm3 3cm 8cm 8cm 3 54 cm3 2cm 4 8cm 3c 183.3cm3 7cm 5cm 9cm Menu © Boardworks Ltd 2001