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1.2<strong>Volume</strong>When designing a building, it is important for the architect to measurecarefully so that the structure functions properly and is pleasing to theeye. Look at the buildings in the photograph. What geometric shapes canyou identify? What problems could arise if incorrect measurements areused in constructing these shapes?sphere cylinder rectangular prism triangular prismResearch the Internet for at least three cities or places with buildings orstructures that have geometric shapes. Provide an image of each structureand indicate what geometric shapes are used. Which structures are madeup of two or more different three-dimensional figures? What threedimensionalfigures are used to make up each structure?18 MHR • Chapter 101_FFCM12_CH1.indd 183/6/09 12:17:22 PM
Example 1<strong>Volume</strong> of a PrismDetermine the volume of concrete needed to construct the rampshown, to the nearest tenth of a cubic metre.1.2 m2.5 m45.0 cmLiteracyConnectThe base of a prismrefers to one of thetwo congruent,parallel polygonsides. Dependingon the orientationof the prism, thebases can be on thetop and bottom,the front and back,or the left and rightsides.SolutionThe ramp is in the shape of a triangular-based prism.Determine the area of the triangular base.45.0 cm2.5 mA = _ 1 (b × h)2= _ 1 (2.5 × 0.45)45 cm = 0.45 m2= 0.5625The base area of the prism is 0.5625 m 2 . Multiply this value by thedepth of the prism to determine its volume.1.2 m45.0 cm0.5625 m 2<strong>Volume</strong> = Base area × Depth= 0.5625 × 1.2= 0.675Approximately 0.7 m 3 of concrete is needed to construct the ramp.1.2 <strong>Volume</strong> • MHR 1901_FFCM12_CH1.indd 193/6/09 12:17:23 PM
Example 2<strong>Volume</strong> of a CylinderA cylindrical container is to beplanted with flowers. The containeris be 2 ft high, and must hold 1.5 m 3of soil. What is the minimumdiameter, to the nearest centimetre?SolutionConvert the height to metres.1 ft = 0.3048 m2 × 0.3048 = 0.6096The height is 0.6096 m.Substitute h = 0.6096 and V = 1.5 intothe formula for volume of a cylinderand solve for the radius.Method 1: Substitute, then rearrange.V = πr 2 h1.5 = πr 2 (0.6096)1.5 = 0.6096πr 2_ 1.50.6096π = _ 0.6096πr2Divide both sides by 0.6096π.0.6096π_____0.783 r 2 __√ 0.783 = √ r 2Take the square root of both sides.0.885 rMethod 2: Rearrange, then substitute.V = πr 2 hV_πh = _ πr2 hπh√ ___ __V_πh = √ r 2r = √ ___V_πhr = √ ______ 1.5π(2)r 0.885V = 1.5 m 3Divide both sides by πh.Take the square root of both sides.Substitute the values of V and h.Determine the minimum diameter.d = 2r= 2(0.885)= 1.77The minimum diameter of the container is 1.77 m or 177 cm.h = 2 ft20 MHR • Chapter 101_FFCM12_CH1.indd 203/6/09 12:17:23 PM
Example 3<strong>Volume</strong> of a Composite FigureDetermine the volume, to two decimal places, of concrete needed toconstruct this staircase.0.9 m45 cmSolutionThis staircase is in the shape of aprism with a staircase-shaped base.Determine the area of the base andthen multiply by the depth.Method 1: Use components.The base consists of sixcongruent squares.6 × 15 2 = 1350The total base area is 1350 cm 2 .Multiply the base area by the depth todetermine the volume of the prism.0.9 m1350 cm 2<strong>Volume</strong> = Base area × Depth= 0.1350 × 0.9= 0.243Convert the base area to square metres:1350 cm 2 ÷ (100 × 100) = 0.1350 m 2 .The volume of concrete needed to construct the staircase isapproximately 0.24 m 3 .1.2 <strong>Volume</strong> • MHR 2101_FFCM12_CH1.indd 213/6/09 12:17:24 PM
Method 2: Use the volume of a simpler object.Think of the staircase as part of a largersquare-based prism.The staircase consists of six of the nine congruentrows of cubes shown. The volume of the staircaseis _ 6 9 or _ 2 the volume of the square-based prism.3<strong>Volume</strong> = _ 2 (Base area × Depth)3= _ 2 3 (0.452 × 1.8)= _ 2 3 (0.3645)= 0.243Convert 45 cm to0.45 m and substitute.The volume of concrete needed to construct the staircase is 0.24 m 3 .Key Concepts• To apply a volume formula, all measures must be in common units.• The volume of a prism, V, can be calculated by multiplying its basearea by its height or depth.V = Base Area × Depth or V = Base Area × Height• The volume of a cylinder can be calculated using the formula V = πr 2 h.Discuss the ConceptsD1. a) What distinguishes a prism from other three-dimensionalobjects?b) Sketch three different prisms and identify them by theshapes of their bases.D2. a) Can a cylinder be thought of as a prism? Explain.b) Explain how the formula for the volume of a cylindercan be found by applying the relationship:V = Base area × Depth or HeightD3. Describe the steps12 myou would follow tocalculate the volumeof this storage tank.15 ft22 MHR • Chapter 101_FFCM12_CH1.indd 223/6/09 12:17:24 PM
PractiseA1. a) Identify the shape of the base of this prism.b) The base area of the prism is 25 cm 2 and its height is 4 cm.Determine the volume of the prism.2. a) Which units would youuse for the volume of thissquare-based prism?Explain.b) Calculate the volume ofthe prism.12 cm25 mm3. Refer to the prism in question 2. Could this prism be describedas a rectangular-based prism? Explain, using words and diagrams.Reasoning and ProvingRepresenting Selecting ToolsProblem SolvingConnecting ReflectingCommunicating4. A cylinder has diameter 6.2 cm and height 11.4 cm.a) Sketch and label a diagram of the cylinder.b) Determine the volume of the cylinder to the nearestcubic centimetre.c) What volume of liquid will this cylinder hold, to two decimalplaces? Recall 1 L = 1000 cm 3 .ApplyB5. Determine the volume of Fido’s doghouse.1.5 mFido2.0 m1.0 m1.4 m1.2 <strong>Volume</strong> • MHR 2301_FFCM12_CH1.indd 233/6/09 12:17:25 PM
10. a) Identify an object in your classroom or in a room at homethat is in the shape of a prism. Sketch the object and label itwith its measures.b) Determine the volume of the object.11. Repeat question 10 for a prism with a base that is acomposite figure.ExtendC12. Airplane fuel tanks are often located in the aircraft’s wings. Tofit inside the wing without any wasted space, some fuel tanks aredesigned in the shape of cylindrical prisms whose ends are ovals(ellipses) rather than circles.badThe formula for the area of an ellipse is A = πab. The wing of anaircraft can accommodate an ellipse with a = 120 cm and b = 50 cm.a) Determine the depth, d, of the tank if it has to hold 200 L of fuel.b) How is the formula for the area of an ellipse related to the formulafor the area of a circle? Explain.13. Charlene is reading a book that has a volume of 120 in. 3 , and coverdimensions of 8 in. by 10 in., as shown.The front andback covers areeach 0.5 cm thickand each page is0.05 mm thick.Charlene can read20 pages per hour.Determine howmany hours it willtake for Charleneto read the entirebook.8 in.10 in.1.2 <strong>Volume</strong> • MHR 2501_FFCM12_CH1.indd 253/6/09 12:17:26 PM
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