S8.6 Volume. Section 1. Surface area of cuboids: Q1. Work out the surface area of each cuboid shown below:

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1 Things to Learn (Key words, Notation & Formulae) Complete from your notes Radius- Diameter- Surface Area- Volume- Capacity- Prism- Cross-section- Surface area of a prism- Surface area of a cylinder- Volume of a prism- Section 1. Surface area of cuboids: Q1. Work out the surface area of each cuboid shown below: Q2. What is the surface area of a cuboid with the dimensions 4cm, 5cm and 6cm? Section 2. Volume of cuboids: Q1. Calculate the volume of these cuboids:

2 Q2. Section 3. Definition of prisms: Label all the shapes and tick the ones that are prisms. Section 4. Surface area of prisms: Q1. Find the surface area of this triangular prism. Q2. Calculate the surface area of this cylinder.

3 Q3. Cans are in cylindrical shapes. Each can has a diameter of 5.3 cm and a height of 11.4 cm. How much paper is required to make the label for the 20 cans? Section 5. Volume of a prism: Q1. Find the volume of this L-shaped prism. Q2. Calculate the volume of this prism. Give your answer to 2sf Section 6. Volume of a cylinder: Q1. Calculate the volume of this cylinder.

4 Q2. The diagram shows a piece of wood. The piece of wood is a prism of length 350cm. The cross-section of the prism is a semi-circle with diameter 1.2cm. Calculate the volume of the piece of wood. Give your answer to 3sf. Section 7. Problems involving volume and capacity: Q1. Sam buys a planter shown below. The planter is 2.5m wide, 3m long and 1.5m deep. How much soil does Sam need to fill the planter 75cm deep? Q2. The cuboid shown below has a volume of 60 cm 3. The base is 6.2 cm long and 3.7 cm wide. Not to scale 3.7 cm 6.2 cm Calculate the height of the cuboid.

5 Q3. The triangular base of a prism is a right triangle with sides a and b= 2a. The height h of the prism is 10mm and its volume is equal to 40mm 3. Find the length of the sides a and b. Section 8. Converting units: Q1. Consider a cuboid that is 1cm by 1cm by 1cm. It has a volume of 1cm 3. What is the volume in mm 3? Q2. Convert a volume of 382,000 cm 3 to m 3. Q3. A swimming pool measures 3m deep and has a base with area 375m 2. a) Find the volume of the pool in cm 3. b) How many litres of water can the pool hold?

6 Answers Section 1. Surface area of cuboids: Q1. Surface area = 2 x Area of the front faces + Perimeter of the front face x length Surface area of 1 st cuboid= 2( 10 x 6) + 2(10 x 3) + 2(6 x 3) face is a rectangle) = = 216cm 2 ( Each individual Or Using 2A + PL = 2 (3 x 6) + 10 ( ) = 216cm 2 Surface area of 2 nd cuboid= 2( 5 x 11) + 2 ( 2 x 5) + 2 ( 11 x 2) = = 174cm 2 Q2. Opposite faces of a cuboid have same area, so we will double each area of the rectangular face and then add them up to get the total surface area. Therefore, Surface area = 2 ( 4 x 5) + 2( 5 x 6) + 2 ( 6 x 4) = = 148cm 2 Or Surface area = 2A + PL = 2 x (4x 5) + 6 x ( ) = = 148cm 2 Section 2. Volume of cuboids: Q1. Volume of a cuboid = Length x Width x height Volume of 1 st cuboid = 2 x 6 x 4 = 48cm 3 Volume of 2 nd cuboid = 5 x 5 x 2 = 50cm 3

7 Q2. Let the width of the cuboid be W. Since the volume = L x W x H Therefore, 8 x W x 3 = W = 120 W = 120/24= 5 cm Section 3. Definition of prisms:- A 3-D Polyhedron (A 3-D shape that has straight edges and flat faces) with a constant cross-sectional area. Cuboid (Prism) Cylinder (Not a prism) Triangular Prism Sphere (Not a prism) Hemi-sphere (Not a prism) Cone (Not a prism) Hexagonal Prism Tetrahedron(Not a prism) Cuboid (Prism) Section 4. Surface area of prisms: Q1. Area of the front and the back face = 2 x Area of each triangle = 2 x ½ x base x perpendicular height = base x perpendicular height = 6 x 8 = 48units 2 Perimeter of triangle = = 26 units Surface Area = 2A +PL= x 26 = 360units 2

8 Q2. Surface area of the cylinder = Area of the top circle + area of the base circle + area of the curved part Area of the top circle = Area of the base = area of the circle with radius (r) of 4cm =πr 2 = π(4) 2 = 16 π cm 2 Area of the curved part = Area of the rectangle with length of 6cm and width same as the circumference of the circular parts. Circumference of the circular parts = 2 πr = 2 π(4) = 8 π Therefore, area of the curved part = 6 x 8 π = 48 π cm 2 Hence, Surface area of the cylinder= 16 π + 16 π + 48 π = 80 π = = cm 2 (2 d.p.) Q3. Since the labels only cover the curved part of the cylinders, we only need to find the area of the curved parts (rectangles). Radius (r) = ½ of diameter = ½ x 5.3 = 2.65 cm Height of the cylinder= length of the rectangular part= 11.4 cm Width of the rectangle = Circumference of the circle = 2πr = 2π 2.65 = Hence, Area of one label= 11.4 x = cm 2 Area of the paper required to make labels for 20 cans = 20 x = cm 2 = cm 2 (1 d.p) Section 5. Volume of a prism: Q1. Volume of prism = Area of cross-section x length of prism Length of this prism = 10mm Area of the cross- section of this prism= area of the front face (L-shaped)

9 = Area of the blue vertical rectangle + Area of the white horizontal rectangle = (20 x 60) + ( 20 x 60) = = 1400mm 2 Volume of the prism = 1400 x 10 = 14000mm 3 Q2. Area of a semi-circle = ½ of the area of a circle Semi-circle with radius 30cm Length of the prism Area of the cross-section= Area of the semi-circle + area of the rectangle = ½ x π x(30) 2 + (60 x 45) = ½ x π x = cm 2 Therefore, Volume of the prism = x 90 = cm 2 = cm 3 (3 s.f.)

10 Section 6. Volume of a cylinder: Q1. Volume of a cylinder = πr 2 h Radius (r) = 4cm, Height (h) = 8cm Therefore, Volume = π(4) 2 (8) = π 16 8 = 128π Q2. The piece of wood is half of a cylinder. = cm 3 = cm 3 (3dp) Volume = ½ of the volume of the total cylinder Radius (r) = ½ x 1.2= 0.6 cm Height (h) = 350cm Volume = ½ x πr 2 h = ½ x π(0.6) 2 (350) = 1/2 x π x 0.36 x 350 = cm 3 = 198cm 3 (3 sf) Section 7. Problems involving volume and capacity: Q1. The planter is in the shape of a cuboid. Length = 3m, width = 2.5 m If the depth is 75cm = 0.75m (1cm = 1/100m) Volume of soil required = 3 x 2.5 x 0.75 = 5.625m 3 Q2. Volume = length x width x height Length = 6.2cm, width = 3.7cm, Volume = 60cm 3 Therefore, 60 = 6.2 x 3.7 x height 60 = x height Height = 60 / = = 2.62cm (2dp) Q2. Area of the cross-section= Area of the right angled triangle = ½ x base x height = ½ x a x b = ½ x a x 2a (Substituting b= 2a)

11 = a 2 Length of the prism = 10mm and volume = 40mm 3 Therefore, 40 = 10 x a 2 (Volume = area of cross-section x length) 4 = a 2 (dividing by 10 both sides) a = 2mm (Square rooting both sides) Since b= 2a, b = 2 x2 = 4mm Section 8. Converting units: Q1. 1cm = 10mm 1cm 3 = 1cm x 1cm x 1cm= 10mm x 10mm x 10mm = 1000mm 3 Q2. 1cm = 1/ 100 m 382,000cm 3 = x 1 = 382 x = 0.382m3 x 1 x m3 Q3. a) Volume of the pool = depth x area of the base = 3 x 375 = 1125m 3 b) 1litres = 1000 cm 3, 1cm 3 = 1/1000 litres 1125m 3 = 1125 x 100 x 100 x100 cm 3 = / 1000 litres = litres