Surface Area

MATH LINKCity PlanningWhen city planners design communities, they consider thepurpose of the buildings, the width of the streets, theplacement of street signs, the design and placement oflampposts, and many other items found in a city.Communication and cooperation are keys to being successful,because city planners have to coordinate and work with manyother people.Imagine that you are a city planner for a miniature community.Discuss your answers to #1 and #2 with a partner, then sharewith your class.1. a) What buildings are essential to a new community?b) What different shapes are the faces of these buildings?2. What other items are important to include in a community?3. Using grid paper, sketch all or part of an aerial view of acommunity including the essential buildings your classdiscussed. Make sure to include roads and any otherfeatures that are important.In this chapter, youwill work in groups tocreate and design aminiature community.Math Link • MHR 5

Views of Three-DimensionalObjectsFocus on…After this lesson,you will be able to... draw and labeltop, front, andside views of3-D objects build 3-Dobjects whengiven top, front,and side views• 20 unit blocks• masking tape• isometric dot paperTo describe athree-dimensional(3-D) object, count itsfaces, edges, andvertices.Face:flat orcurvedsurfaceVertex: pointwhere three ormore edges meetEdge: linesegmentwhere twofaces meetSable and Josh are trying to build exactly the same three-dimensional(3-D) object. They each have the same number of blocks, but theycannot see each other’s object.Using a common vocabulary can help Sable and Josh build thesame object.How can you describe and build three-dimensional objects?1. Work with a partner. Create a 3-D object using ten unit blocks.Make sure your partner cannot see your object.2. Describe your completed object to your partner, and have yourpartner try to build the same object. What key words did youuse that were helpful?3. Decide which faces will be the front and top of your object. Thendetermine which faces are the bottom, left side, right side, and back.You may wish to label the faces with tape. Then, describe yourobject to your partner again. Was it easier to describe this time?6 MHR • Chapter 5

4. Using isometric dot paper, draw what your object looks like.Reflect on Your Findings5. a) Do you need to know all the views to construct an object?If not, which ones would you use and why?b) Explain why you might need to have only one side view,if the top and front views are also given.c) Are any other views unnecessary? Are they needed toconstruct the same object?Using isometric dot paper makesit easier to draw 3-D shapes.Follow the steps to draw arectangular solid.13 42Each view shows twodimensions. Whencombined, these viewscreate a 3-D diagram.Example 1: Draw and Label Top, Front, and Side ViewsUsing blank paper, draw the top, front, and side views of these items.Label each view.a) Tissue box b) Compact disk caseSolutiona) top front side(end of the box)b) top front side5.1 Views of Three-Dimensional Objects • MHR 7

Using blank paper, draw the top, front, andside views of this object.Architects use topviews to drawblueprints forbuildings.Example 2: Sketch a Three-Dimensional Object When Given ViewsThese views were drawn for an object made of ten blocks.Sketch what the object looks like.topfrontsideSolutionUse isometric dot paperto sketch the object.An object is created using eightblocks. It has the following top,front, and side views. Sketch whatthe object looks like on isometricdot paper.topfrontside8 MHR • Chapter 5

Example 3: Predict and Draw the Top, Front, and Side ViewsAfter a RotationThe diagrams show the top, front, and side views of the computer tower.top front sideYou want to rotate the computer tower 90º clockwise on its base to fitinto your new desk. Predict which view you believe will become thefront view after the rotation. Then, draw the top, front, and side viewsafter rotating the tower.This diagram showsa 90° clockwiserotation.90°SolutionThe original side view will become the new front view after the rotation.top front sideYou can use a Drawprogram to create3-D objects.Stand your MathLinks 8 textbook on your desk. Predict what the top,front, and side views will look like if you rotate it 90º clockwise aboutits spine. Then, draw the top, front, and side views after rotatingthe textbook.5.1 Views of Three-Dimensional Objects • MHR 9

• A minimum of three views are needed to describe a 3-D object.• Using the top, front, and side views, you can build or draw a3-D object.topfrontside1. Raina insists that you need to tell her all six views so she can drawyour object. Is she correct? Explain why or why not.2. Are these views correct? Justify your answer.fronttopsidec)For help with #3 and #4, refer to Example 1 onpage XXX.3. Sketch and label the top, front, andside views.a) b)PhotoAlbum4. Choose the correct top, front, and sideview for this object and label each one.ABCD10 MHR • Chapter 5EFG

For help with #5, refer to Example 2 on page xxx.5. Draw each 3-D object using theviews below.a) topfront side8. Choose two 3-D objects from yourclassroom. Sketch the top, front, andside views for each one.b) top front side9. Sketch the front, top, and right sideviews for these solids.a)b)For help with #6 and #7, refer to Example 3 onpage xxx.6. A television set has the following views.c)frontfronttopfrontsidefrontIf you turn the television 90°counterclockwise, how would the threeviews change? Sketch and label eachnew view.7. Choose which objecthas a front view like thisafter a rotation of90º clockwise onto its side.a)set of booksb)CD rack10. Describe two objects that meet thisrequirement: When you rotate one object90 degrees, the top, front, and side views arethe same as the top, front, and side views ofthe other object that was not rotated.11. An injured bumblebee sits at a vertex of acube with edge length 1 m. The bee movesalong the edges of the cube and comesback to the original vertex without visitingany other vertex twice.a) Draw diagrams to show thebumblebee’s trip around the cube.b) What is the length, in metres, of thelongest trip?MATH LINKChoose one of the essential buildings that you discussed for your new communityon page XXX. Draw and label a front, side, and top view.5.1 Views of Three-Dimensional Objects • MHR 11

2. Predict which nets will form a right rectangular prism.3. Cut each net out along the outside edges and fold along the insideedges, taping the cut edges to try to form a right rectangular prism.4. Do all the nets create right rectangular prisms?5. Place a right rectangular prism (such as a small cardboard box) ona piece of blank paper. “Roll” the prism onto its faces, trace eachface, and try to draw another correct net. Your net should bedifferent from the examples you have already made.Reflect on Your Findings6. a) Compare the net you drew with those of three of yourclassmates. What is the same and different about your nets?b) Is there more than one way to draw a net for a 3-D object?Explain your answer.Example 1: Draw a Net for a Three-Dimensional ObjectA company asks you to create an umbrella standfor large beach umbrellas. Draw the net for theumbrella stand.SolutionVisualize what the umbrella stand would look like if you could cut itopen and flatten it. The net has one circle and a rectangle. When therectangle is curved around the circle, the net will form a cylinder withan open top. The width of the rectangle is equal to the circumferenceof the circle.StrategiesChange Your Pointof ViewRefer to page xxx.Draw a net for an unopened soup can.5.2 Nets of Three-Dimensional Objects • MHR 13

Example 2: Build a Three-Dimensional Object From a Given NetBefore going to leadership camp, your group needs to put a tenttogether. Can this net be folded to form the shape of a tent?StrategiesModel ItRefer to page xxx.triangular prism• a prism with twotriangular bases eachthe same size andshapeSolutionTrace the net onto paper. Cut along the outside edges and foldalong the inside edges. Tape the cut edges together to try to builda right triangular prism .The net can be folded to form the shape of a tent.Build a 3-D object using this net. What objectdoes it make?14 MHR • Chapter 5

• A net is a two-dimensional shape that, when folded,encloses a three-dimensional object.• The same 3-D object can be created by foldingdifferent nets.netcube• You can draw a net for an object by visualizing what it wouldlook like if you cut along the edges and flattened it out.1. Both of these nets have six faces, like a cube. Will both netsform a cube? Justify your answer.net Anet B2. Patricia is playing the lead role in the school musical this year. Shemissed Math class while she was performing. She cannot figure outif a net will build the correct 3-D object, and asks you for help afterschool. Show how you would help her figure out this problem.For help with #3 to #5, refer to Example 1 on page xxx.3. Sketch a net for each object.a)b)c)hockey puckchocolate barjewellery box5.2 Nets of Three-Dimensional Objects • MHR 15

4. Draw the net for each object. Label themeasurements on the net.a) d = 30 mm7. Match each solid with its net. Copy thenets, then try to create the 3-D objects.78 mmA ream describes aquantity ofapproximately 500sheets of paper.rectangular prismcylinderb)triangular prismAB28 cmPaper500 Sheets21.5 cm5 cm5. Draw a net on grid paper for a rectangularprism with the following measurements:length is six units, width is four units, andheight is two units.CFor help with #6 and #7, refer to Example 2 onpage xxx.6. a) Draw the net on grid paper, as shown.Cut along the outside edges of the netand fold to form a 3-D object.DEb) What is this object called?8. A box of pens measures 15.5 cm by 7 cmby 2.5 cm. Draw a net for the box on apiece of centimetre grid paper. Then, cutit out and fold it to form the box.9. You are designing a new mailbox. Drawa net of your creation. Include allmeasurements.16 MHR • Chapter 5

10. Angela designed two nets.12. The six sides of a cube are each a differentcolour. Four of the views are shown below.a) Enlarge both nets on grid paper, andbuild the 3-D objects they form.b) What object does each net form?11. Hannah and Dakota design a spellingboard game. They use letter tiles to createwords. Tiles may be stacked (limit of four)on top of letters already used for a wordon the board to form a new word.a) Draw a 3-D picture of what thesestacked tiles might look like.b) Draw a top view that illustrates thestacked tiles for people reading theinstructions.What colour is on the opposite side of eachof these faces?a) purpleb) bluec) red13. How many possible nets can create a cube?Sketch all of them. The first one is donefor you.MATH LINKWhen buildings are designed, it isimportant to consider engineeringprinciples, maximum andminimum height requirements,and budget.a) Create a 3-D sketch of twobuildings for your miniaturecommunity, one that is a prismand one that is a cylinder.b) Draw a net of each building,including all possiblemeasurements needed tobuild your miniature.5.2 Nets of Three-Dimensional Objects • MHR 17

Surface Area of a PrismFocus on…After this lesson,you will be able to... link area tosurface area find the surfacearea of a rightprismMost products come in some sort of packaging. You can helpconserve energy and natural resources by purchasing products that• are made using recycled material• use recycled material for packaging• do not use any packagingWhat other ways could you reduce packaging?• empty cardboard box(cereal box, granolabox, snack box, etc.)• scissors• ruler• scrap paperHow much cardboard is needed to make a package?1. Choose an empty cardboard box.a) How many faces does your box have?b) Identify the shape of each face.2. Cut along the edges of the box and unfold it to form a net.Literacy LinkExcludeThe dimensions of anobject are measuressuch as length,width, and height.18 MHR • Chapter 53. What are the dimensions and area of each face?overlappingflaps.4. How can you find the amount of material used to make yourcardboard box?

Reflect on Your Findings5. a) How did you use the area of each face to calculate the totalamount of material used to make your cardboard box?b) Can you think of a shorter way to find the total area withouthaving to find the area of each face? Explain your method.Example 1: Calculate the Surface Area of a Right Rectangular Prisma) Draw the net of this right rectangular prism.b) What is the area of each face?c) What is the surface area of the prism?Solutiona)4 cm10 cm6 cm6 cm10 cm4 cmsurface area• the number of squareunits needed to covera 3-D object• the sum of the areas ofall the faces of anobjectb) The right rectangular prism has faces that are three different sizes.6 cmfront or back10 cmA = l × wA = 10 × 6A = 60The area of the frontor back is 60 cm 2 .4 cmtop or bottom10 cmA = l × wA = 10 × 4A = 40The area of the topor bottom is 40 cm 2 .4 cmends6 cmA = l × wA = 6 × 4A = 24The area of eachend is 24 cm 2 .Area is measured insquare units.For example, squarecentimetres, squaremetres, etc.c) The total surface area is the sum of the areas of all the faces.The front and backhave the same area:A = 60 × 2A = 120The top and bottomhave the same area:A = 40 × 2A = 80The two ends havethe same area:A = 24 × 2A = 48Total surface area = (area of front and back) + (area of top and bottom)+ (area of ends)= 120 + 80 + 48= 248The surface area of the right rectangular prismis 248 cm 2 .You could add the areas you calculatedfirst. 60 + 40 + 24 = 124Each area is the same as the area of one otherface, so you could then multiply the total bytwo. 124 × 2 = 2485.3 Surface Area of a Prism • MHR 19

What is the surface area of this rightrectangular prism?16 cm8 cm3 cmExample 2: Calculate the Surface Area of a Right TriangularPrisma) Draw the net of this right triangular prism.b) What is the area of each face?c) What is the total surface area?Solution9 m2.6 ma)3 mStrategiesDraw a DiagramRefer to page xxx.3 m2.6 m9 mLiteracyLinkAn equilateraltriangle has threeequal sides and threeequal angles. Equalsides are shown ondiagrams by placingtick marks on them.b) The bases of the prism are equilateral triangles.The sides of the prism are rectangles.3 mrectangletriangle9 m 3 m2.6 mA = l × w A = (b × h) ÷ 2A = 9 × 3 A = (3 × 2.6) ÷ 2A = 27 A = 7.8 ÷ 2A = 3.9The area of oneThe area of onerectangle is 27 m 2 . triangle is 3.9 m 2 .20 MHR • Chapter 5

c) You must add the areas of all faces to find the surface area.This right triangular prism has five faces.There are three rectangles of the same size and two trianglesof the same size.Total surface area = (3 × area of rectangle) + (2 × area of triangle)= (3 × 27) + (2 × 3.9)= 81 + 7.8= 88.8The surface area of the right triangular prism is 88.8 m 2 .Find the surface area of thistriangular prism.7 cm9.9 cm7 cm2 cm• Surface area is the sum of the areas of all the faces of a 3-D object.A1A6A2A3A4A5Surface Area = A1 + A2 + A3 + A4 + A5 + A6,where A1 represents the area of rectangle 1, A2represents the area of rectangle 2, etc.1. Write a set of guidelines that you could use to find the surface areaof a prism. Share your guidelines with a classmate.2. A right rectangular prism has six faces. Why do you have to findthe area of only three of the faces to be able to find the surfacearea. Use pictures and words to explain your thinking.5.3 Surface Area of a Prism • MHR 21

For help with #3 and #4, refer to Example 1 onpage xxx.3. Find the surfacearea of this rightrectangular prism tothe nearest tenth of asquare centimetre.18.5 cm13.5 cm5 cm7. Given the area of each face of a rightrectangular prism, what is the surface area?fronttopside20 mm 2 12 mm 2 15 mm 28. Paco builds a glass greenhouse.4. Find the surface areaof this CD case.14 cmFor help with #5 to #7, refer to Example 2 onpage xxx.12.3 cm 1 cm5. Calculate thesurface area of2.7 mthis ramp in the1.5 mshape of a righttriangular prism.2.3 mGive your answerto the nearest tenth of a square metre.0.7 m1.1 m1.8 m0.6 m2.4 ma) How many glass faces does thegreenhouse have?b) How much glass does Paco need to buy?9. What is the minimum amount of materialneeded to make the cover of this textbookif there is no overlap? Give your answer tothe nearest square millimetre.10. Jay wants to make a bike ramp. He drawsthe following sketch. What is the surfacearea of the ramp?6. Cheese is sometimes packaged in atriangular box. How much cardboardwould you need to cover this piece ofcheese if you do not include overlapping?Calculate your answer to the nearest tenthof a square centimetre.0.9 m1.6 m2.2 m2 m4.5 cm3 cm9 cm6.4 cmThe tick marks onthe two sides ofthe triangleindicate that thesesides are equal.22 MHR • Chapter 5

11. Dallas wants to paint three cubes. Thecubes measure 1 m × 1 m × 1 m,2 m × 2 m × 2 m, and 3 m × 3 m × 3 m,respectively. What total surface area willDallas paint if he decides not to paint thebottoms of the three cubes?12. Tadika has a gift to wrap. Both of thesecontainers will hold her gift. Whichcontainer would allow her to use theleast amount of wrapping paper? Explainyour choice.5 cm10 cm7 cm30 cm10 cm5 cm13. A square cake pan measures 30 cm oneach side and is 5 cm deep. Cody wants tocoat the inside of the pan with non-stickoil. If a single can of non-stick oil coversan area of 400 000 cm 2 , how many panscan be coated with a single can?14. Ethan is hosting games night this weekend.He bought ten packages of playing cards.Each package measures 9 cm × 6.5 cm ×1.7 cm. He wants to build a container tohold all ten packages of cards.a) What are the minimum insidedimensions of the container?b) Is there more than one kind ofcontainer that would work? Drawdiagrams to help explain your answer.15. a) If the edge length of a cube is doubled,find the ratio of the old surface area tothe new surface area.b) What happens if the edge length ofa cube is tripled? Is there a pattern?16. Shelby wantsto paint thewalls andceiling of arectangularroom.2.6 m 6.8 m4.8 mType of Paint Size of Paint Can CostWall paint4 L1 L$24.95$7.99Ceiling paint 4 L $32.95One litre of paint covers 9.5 m 2 .a) What is the least amount of paint Shelbycan buy to paint the room (subtract 5 m 2for the door and windows)?b) How much will the paint cost,including the amount of tax chargedin your region?MATH LINKFor the prism-shaped building you created in the Math Link on page XXX, how muchmaterial do you need to cover the exterior walls and the roof of the building?5.3 Surface Area of a Prism • MHR 23

Surface Area of a CylinderFocus on…After this lesson,you will be able to... find the surfacearea of a cylinderGlow sticks work because of a chemical reaction. There are twosolutions in separate compartments inside the stick. Once you bend thestick, the two solutions mix. This mixture creates a new solution thatgives off light. The colour of the glow stick depends on the dye in themixture. How might you determine how much plastic would be neededto make a glow stick to fit around your wrist?cylinder• a three-dimensionalobject with twoparallel and congruentcircular basesHow do you find the surface area of a right cylinder ?1. Draw the net of a glow stick. Use theactual dimensions from the diagramshown.21 cmd = 0.5 cmcylinderIf your calculator hasa π key, you can useit to get a moreaccurate answer.2. List the shapes that make up your net.3. a) Copy and complete the table.Area Formula MeasurementsCircle: A = × π = 3.14 r = Rectangle: A = × l = w = Use 3.14 as anapproximate value for π.The radius of a circle isone half the diameter.b) What measurement are you missing to calculate the area of eachshape? Use your piece of paper as a model of a glow stick to helpyou visualize what might be missing.c) How can you use what you know about circles to help you findthe missing measurement?24 MHR • Chapter 5

4. a) The cylinder has two identical circles, one at each end.What is the area of each circle?b) What is the area of the rectangle?c) The total surface area is the sum of the areas of all the shapes.What is the surface area of the glow stick? Include the units inyour final answer.Pop cans arecylinders. The world’slargest Coke can islocated in Portage laPrairie, Manitoba.Reflect on Your Findings5. a) How would you find the surface area of any right cylinder?b) What type of units do you use to measure surface area?Example 1: Determine the Surface Area of a Right Cylindera) Estimate the surface area of the can.b) What is the surface area of the can? Expressyour answer to the nearest hundredth of asquare centimetre?11 cmSolution7.5 cmThe surface area of the can is found by adding the areas of the twocircular bases and the rectangular side that surrounds them.The width, w, of the rectangle is the height of the can.The length, l, of the rectangle is equal to the circumference of the circle.a) To estimate, use approximate values: d ≈ 8 cm, w ≈ 10 cm, π ≈ 3.Area of circle = π × r 2≈ 3 × 4 × 4≈ 48There are two circles: 2 × 48 = 96The area of the two circles is approximately 96 cm 2 .Area of rectangle = l × w= (π × d) × w Replace l with the formula for the≈ 3 × 8 × 10 circumference of a circle.≈ 240The area of the rectangle is approximately 240 cm 2 .Estimated surface area = area of two circles + area of rectangle≈ 96 + 240≈ 340The estimated surface area is 340 cm 2 .r 2 meansr × rThe radius of a circleis half the diameter.LiteracyLinkcircleradiuscentrediameterThe formula for thecircumference of a circle isC = π × d or C = 2 × π × r.5.4 Surface Area of a Cylinder • MHR 25

StrategiesDraw a DiagramRefer to page xxx.b) Method 1: Use a NetDraw the net and label the measurements.top side bottom7.5 cm11 cmThe diameter of the circle is 7.5 cm.Determine the radius.7.5 ÷ 2 = 3.75The radius of the circle is 3.75 cm.Round youranswer at the end ofthe calculation.Find the area of one circle.A = π × r 2Use 3.14 as anapproximate valueA ≈ 3.14 × 3.75 2for π.A ≈ 44.15625The area of one circle is approximately 44.15625 cm 2 .Find the area of two circles.2 ×44.15625 = 88.3125The area of both circles is approximately 88.3125 cm 2 .Find the area of the rectangle using the circumference of the circle.A = l × wA = (π × d) × w Replace l with the formula for the circumference of a circle.A ≈ 3.14 × 7.5 × 11A ≈ 259.05The area of the rectangle is approximately 259.05 cm 2 .Calculate the total surface area.Total surface area = 88.3125 + 259.05= 347.3625The total surface area is approximately 347.36 cm 2 .26 MHR • Chapter 5

Method 2: Use a Formula.Use this formula to find the total surface area of any cylinder.S.A. = 2 × (π × r 2 ) + (π × d × h)S.A. ≈ 2 × (3.14 × 3.752) + (3.14 × 7.5 × 11)S.A. ≈ 88.3125 + 259.05S.A. ≈ 347.3625The total surface area is 347.36 cm 2 , to thenearest hundredth.This formula incorporates each shape and its areaformula to find the surface area.2 × (π × r 2 ) + (π × d) × htwo circles circle area rectangle areaformula formula (length is thecircumference of a circle;width is the height ofthe cylinder)Calculate the surface area of thiscylinder to the nearest tenth of asquare centimetre.9 cm55 cmLiteracyLinkThe abbreviation S.A.is often used as ashort form for surfacearea.Example 2: Use the Surface Area of a CylinderCalculate the surface area of this totem pole, including the two circularbases. The pole stands 2.4 m tall and has a diameter of 0.75 m. Giveyour answer to the nearest hundredth of a square metre.SolutionThe cylinder has two circular bases.The area of one circle is:A = π × r 2A ≈ 3.14 × 0.375 2r = d ÷ 2A ≈ 0.4415625The area of the circle isapproximately 0.4415625 m 2 .There are two circles, so the areaof both circles is approximately0.883125 m 2 .Calculate the total surface area.S.A. ≈ 0.883125 + 5.652S.A. ≈ 6.535125The total surface area is approximately 6.54 m 2 .The side of the cylinder isa rectangle.The area of the rectangle is:A = (π × d) × hA ≈ 3.14 × 0.75 × 2.4A ≈ 5.652The area of the rectangle isapproximately 5.652 m 2 .Replace one dimensionwith the formula for thecircumference of a circle.Calculate the surface area of a cylindrical waste backet without a lidthat measures 28 cm high and 18 cm in diameter. Give your answerto the nearest square centimetre.This metal totem polewas created by ToddBaker, Squamish Nation.It represents the Birth ofthe Bear Clan, with theprincess of the clan onthe top half and thebear on the bottom half.5.4 Surface Area of a Cylinder • MHR 27

• The surface area of a cylinder is the sum of theareas of its faces.• A net of a cylinder is made up of onerectangle and two circles.• To find one of the dimensions of the rectangle,calculate the circumference of the circle.The length of this sideis the circumference ofthe circle C = π d orC = 2 π r1. What are the similarities and differences between finding thesurface area of a prism and finding the surface area of a cylinder?2. Explain why you need to find the circumference of a circle to findthe surface area of a cylinder.For help with #3 to #7, refer to Examples 1 and 2 onpages xxx–xxx.3. a) Draw a net for this cylinder.b) Sketch a different net forthis cylinder.4. Estimate the surface area of each cylinder.Then, calculate each surface area to thenearest tenth of a square centimetre.a)d = 7 cm30 cmb)r = 10 cm22 cm5. Find the surface area of each objectto the nearest tenth of a square unit.a)b)d = 2.5 cm16 cmwooden rodflag poled = 0.003 m16 m6. Use the formulaS.A. = 2 × (π × r 2 ) + (π × d × h) tocalculate the surface area of each object.Give each answer to the nearest hundredthof a square unit.a)d = 2.5 cmb)d = 5 cm10 cmYou can simplify the formula:S.A. =2 × (π × r 2 ) + (π × d × h)= 2πr 2 + πdh7 cm28 MHR • Chapter 5

7. Do you prefer to find the surface area ofa cylinder by using the sum of the area ofeach face or by using a formula? Give atleast two reasons for your choice.11. If each tennis ball has a diameter of 7 cm,calculate the amount of material needed tomake a can that holds three tennis balls.8. Anu wants to re-cover the cylindrical stoolin his bedroom. How much material doeshe need if there is no overlap?d = 42 cm32 cm9. Kaitlyn and Hakim each bought a tubeof candy. Both containers cost the sameamount. Which container required moreplastic to make?d = 7 cmd = 11 cmCANDY122 cmCANDY85 cm10. Paper towel is rolledaround a cardboardtube. Calculate theoutside surface areaof the tube.r = 2 cmEach personproduces about1.59 kg of trash eachday. Most of this ispaper products.12. Coins can be stored in a plastic wrappersimilar to a cylinder. A roll of dimescontains 50 coins. Each dime has adiameter of 17.5 mm and a thickness of1 mm. Calculate the minimum surfacearea of the plastic wrapper.13. A paint roller in the shape of a cylinderwith a radius of 4 cm and a length of21 cm is rolled vertically on a wall.a) What is the length and width of thewet path after ten complete rolls?b) What area does the paint cover?27.5 cmMATH LINKFor the cylindrical building you created in the MathLink on page XXX, how much material do you need tocover the exterior walls and the roof of the building?Douglas J. Cardinal, one of the world’s mostacclaimed architects, uses his European, Blackfoot,and Ojibwa roots when designing buildings. He isknown for his design of The Canadian Museum ofCivilization in Gatineau, Québec, as well as anumber of buildings in Western Canada, such asTelus World of Science in Edmonton and FirstNations University of Canada in Regina.5.4 Surface Area of a Cylinder • MHR 29

Key WordsUnscramble the letters for each puzzle in #1 to #6.Use the clues to help you solve the puzzles.1. E T Na flat diagram that you can fold to makea 3-D object2. U S F A R E C E R A Athe sum of the areas of the faces of anobject (2 words)3. I R H T G R P M S Ia prism whose sides are perpendicularto its bases (2 words)4. E C N I Y D R La 3-D object with two parallel circularbases5. I R A G N R U A L T S I M R Pa 3-D object with two parallel triangularbases (2 words)6. L E U C A A N R G T R I R M S Pa 3-D object with two parallel rectangularbases (2 words)5.1 Views of Three-Dimensional Objects,pages xxx–xxx7. Draw and label the top, front, and sideviews for these objects.a) b)8. Using isometric paper, draw each3-D object from the views given.a) front top sideb) front top side9. A filing cabinet is in the far corner of anoffice. Shay is redecorating the room andwants to turn the cabinet 90° clockwise.Here are the views before the turn:front top sidea) How does each view change afterthe turn?b) Draw and label the top, front, andside views of the filing cabinet afterit is turned.5.2 Nets of Three-Dimensional Objects,pages xxx–xxx10. Name the object formed by each net.a) b)30 MHR • Chapter 5

c)15. Find the surface area of each triangularprism.a)6 cm10 cm11. Draw the net for each object.a)b)COOKIESSOUPb)50 cm8 cm22.9 cm20 cm4 cm22.5 cm12. Using two pieces of grid paper, create apencil box and lid. Draw a net, cut it out,fold it, and build your pencil box. Makesure new pencils fit in it!5.3 Surface Area of a Prism, pages xxx–xxxFor #13 to #16, calculate the surface area to thenearest tenth of a square unit.13. What is the surface area of each object?a)16. Liza had two more gifts left to wrap whenshe ran out of paper. Approximately howmuch more wrapping paper does she needto finish wrapping her gifts? Assume nooverlap.32.5 cm35 cm12.5 cm20.5 cm23 cm12.5 cm12 cm12 cm12 cm5.4 Surface Area of a Cylinder,pages xxx–xxxb)For #17 to #19, calculate the surface area tothe nearest tenth of a square unit.0.5 m2 m1.7 m17. Determine the surface 2.5 marea of the cylinder.14. Using the measurements shown on the netof the rectangular prism, calculate thesurface area.10 mm18. The pencil sharpener on Kay’s desk hasa diameter of 3.4 cm and is 7 cm tall.Calculate the surface area.5.5 m27 mm42 mm19. The circumference of a container’s lid is157 cm. If the container is 102 cm tall,what is the surface area of the container?Chapter Review • MHR 31

Short AnswerFor #1 to #5, choose the best answer.C 631.14 cm 2 D 688.01 cm 21. The top view of thiscontainer shows aA circleB squareC triangleD rectangle2. One face on a cube has an area of 49 cm 2 .What is the surface area of the cube?A 343 cm 2B 294 cm 2C 196 cm 2D 154 cm 23. What three-dimensionalobject has a net likethis one?A cubeB cylinderC triangular prismD rectangular prism4. What is the surfacearea of this box?5 mmA 550 mm 220 mmB 900 mm 218 mmC 1100 mm 2D 1800 mm 219 cm5. What is the surface area of a cylinder thatis 30.5 cm long and has a radius of 3 cm,to the nearest hundredth of a squarecentimetre?A 274.50 cm 2 B 603.19 cm 2CRACKERS6. Sketch the top, front, and side viewsof this building.7. An object may have more than one net.Draw three different nets for this cube.8. A DVD case is made of a plastic coveringthat measures 19 cm long, 13.5 cm wide,and 1.4 cm thick. Calculate the surfacearea to the nearest tenth of a squarecentimetre.13.5 cm1.4 cm32 MHR • Chapter 5

9. The surface area of a cube is 1014 cm 2 .Find the length of any side of the cube.Extended Response10. a) Sketch a three-dimensional object youcan build using two of these triangularprisms.12. Single-serving juice boxes measure 10 cmby 7 cm by 5 cm. A manufacturer wantsto shrink wrap four boxes together forsale. Which of the following arrangementsof the boxes will use the least amount ofplastic wrap? Show how you know.10 cmb) Draw the front view, top view, andside view of your object.c) Draw a net for your object.Arrangement 17 cm5 cm11. Ken and Arika are comparing theircylinders. Arika’s cylinder is twice as tallas Ken’s, but is only half the diameter.Ken’s cylinder has a height of 18 cm and adiameter of 9 cm. Whose cylinder has thegreater surface area?7 cmArrangement 210 cm5 cmWRAP IT UP!It is time to create your miniature community!Work together to finalize one aerial view foryour community. You may choose to startwith one that you created on page XXX.Include the following in your diagram anddescription:• All the buildings designed by you and yourgroup members.• A 3-D sketch, net, and surface areacalculations for one new building for eachmember of your group. The new designsshould include at least one prism and cylinder.Check each other’s work before submitting.• Streets to navigate through the city.• Environmental considerations such as watersource, parks, etc.Practice Test • MHR 33

Let’s Face It!1. Play Let’s Face It! with a partner or in a small group. These arethe rules:• Remove the jacks, queens, kings, aces, and jokers from the deckof cards.• Take turns dealing the cards. It does not matter who deals first.• Shuffle the cards and deal three cards, face up, to each player.• Use the values of the cards as the dimensions, in centimetres,of a rectangular prism.• Calculate the surface area of your rectangular prism usingpencil and paper.• Each player who calculates the surface area of their prismcorrectly scores a point. (You will need to check each other’swork.)• The player with the rectangular prism that has the greatestsurface area scores an extra point for that round. If there isa tie, each of the tied players scores an extra point.• The first player to reach ten points wins the game. If more thanone player earns ten points in the same game, these playerscontinue playing until one of them pulls ahead.2. Play a different version of Let’s Face It! by modifying the rulesas follows:• Deal only two cards to each player and use them to describe thesize of a right cylinder. The first card gives the radius of eachcircle, in centimetres. The second card gives the height of thecylinder, in centimetres.• Use a calculator to determine the surface area of your cylinder,to the nearest hundredth of a square centimetre.• Award points and decide the winner in the same way as before.• deck of playing cardsper pair or smallgroup• calculator per studentMy cards are a 5 of clubs,a 3 of hearts, and an 8 ofspades. My rectangularprism has edges of 5 cm,3 cm, and 8 cm.3 cm5 cm8 cmI was dealt a 4 of clubsand then a 6 of clubs.The radius of each circleis 4 cm. The height ofthe cylinder is 6 cm.4 cm6 cm34 MHR • Chapter 5

Design a BedroomHave you ever wondered what itwould be like to completelydesign a room? Suppose you weregiven the opportunity to createthe kind of space that a personyour age would appreciate andmake good use of.You be the interior designer. Yourfirst project is to create a designfor a bedroom that is 4 m wideby 5 m long, and is 2.5 m high.1. a) Draw the top view of theroom and place at leastthree objects in the room.b) Draw the top, front, andside views of at least threeobjects you put in theroom. Identify the 3-Dshape that each objectclosely resembles.2. a) Painting your room is thenext step. Determine theamount of paint you needto cover the walls andceiling of your room.b) One can of the paint youare going to use covers10 m 2 /L. How many cansdo you need?Challenge in Real Life • MHR 35