# Surfa Surf ce ace Area Area What You Will Learn

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1 Surface Area A skyline is a view of the outline of buildings or mountains shown on the horizon. You can see skylines during the day or at night, all over the world. Many cities have beautiful skylines. City planners have to consider much more than just how the skyline will look when they design a city. In the skyline shown in the picture, what shapes do you see? What three-dimensional objects can you identify? In this chapter, you will learn how to draw and build three-dimensional objects and how to calculate their surface areas. What You Will Learn to label and draw views of 3-D objects to draw and build nets for 3-D objects to calculate the surface area for prisms and cylinders to solve problems using surface area 2 MHR Chapter 5

2 Key Words face edge vertex rectangular prism net triangular prism right prism surface area cylinder Literacy Link You can use a Verbal Visual Chart (VVC) to help you learn and remember new terms. Copy the blank VVC into your math journal or notebook and use it for the term, rectangular prism. Write the term in the first box. Draw a diagram in the second box. Define the term in the third box. The glossary on pages XXX XXX may help you. In the fourth box, explain how you will remember the term and what it means. Consider using an example, a characteristic, a memory device, or a visual. Term Definition Diagram How I Will Remember It Chapter 5 MHR 3

3 Making The Foldable sheet of paper ruler glue or tape four sheets of blank paper scissors stapler Step 1 Fold over one of the short sides of an sheet of paper to make a 2.5 cm tab. Fold the remaining portion of paper into quaters. Step 4 Make the paper from Step 3 into eight booklets of 4 pages each. Step 5 Collapse the Foldable. Title the faces of your Foldable. Then, staple the booklets onto each face, as shown, and add the labels shown. 5.1 Views of Three- Dimensional Objects Notes 5.2 Nets of Three- Dimensional Objects What I Need to Work On Wrap It Up! Ideas Notes What I Need to Work On Wrap It Up! Ideas 2.5 cm Step 2 Use glue or tape to put the paper together as shown in the diagram. If you use glue, allow it to dry completely. Step 3 Fold each of four sheets of blank paper into eighths. Trim the edges as shown so that each individual piece is 9.5 cm 6 cm. Cut off all folded edges. 9.5 cm 6 cm Key Words Using the Foldable Key Words As you work through each section of Chapter 5, take notes on the appropriate face of your Foldable. Include information about the examples and Key Ideas in the Notes section. If you need more room, add sheets of paper to your booklet. List and define the key words in the Key Words booklet. Use visuals to help you remember the terms. Keep track of what you need to work on. Check off each item as you deal with it. As you think of ideas for the Wrap It Up!, record them on that section of each face of your Foldable. 4 MHR Chapter 5

4 MATH LINK City Planning When city planners design communities, they consider the purpose of the buildings, the width of the streets, the placement of street signs, the design and placement of lampposts, 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 many other people. Imagine that you are a city planner for a miniature community. Discuss your answers to #1 and #2 with a partner, then share with 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 a community including the essential buildings your class discussed. Make sure to include roads and any other features that are important. In this chapter, you will work in groups to create and design a miniature community. Math Link MHR 5

5 Views of Three-Dimensional Objects Focus on After this lesson, you will be able to... draw and label top, front, and side views of 3-D objects build 3-D objects when given top, front, and side views <Ill 5-1: illustration of two students split view; one girl making an object with unit blocks, one boy making a DIFFERENT object with the same number of unit blocks. Bubbles with dialogue, girl (Sable): I used ten blocks (or how ever many is in the picture), boy (Josh): Sable, what does yours look like if you look at it from the top? > 20 unit blocks masking tape isometric dot paper To describe a three-dimensional (3-D) object, count its faces, edges, and vertices. Face: flat or curved surface Vertex: point where three or more edges meet Edge: line segment where two faces meet Sable and Josh are trying to build exactly the same three-dimensional (3-D) object. They each have the same number of blocks, but they cannot see each other s object. Using a common vocabulary can help Sable and Josh build the same 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 your partner try to build the same object. What key words did you use that were helpful? 3. Decide which faces will be the front and top of your object. Then determine which faces are the bottom, left side, right side, and back. You may wish to label the faces with tape. Then, describe your object to your partner again. Was it easier to describe this time? 6 MHR Chapter 5

6 4. Using isometric dot paper, draw what your object looks like. Reflect on Your Findings 5. 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 to construct the same object? Using isometric dot paper makes it easier to draw 3-D shapes. Follow the steps to draw a rectangular solid Each view shows two dimensions. When combined, these views create a 3-D diagram. Example 1: Draw and Label Top, Front, and Side Views Using blank paper, draw the top, front, and side views of these items. Label each view. a) Tissue box b) Compact disk case Solution a) top front side (end of the box) b) top front side 5.1 Views of Three-Dimensional Objects MHR 7

7 Using blank paper, draw the top, front, and side views of this object. Architects use top views to draw blueprints for buildings. Example 2: Sketch a Three-Dimensional Object When Given Views These views were drawn for an object made of ten blocks. Sketch what the object looks like. top front side Solution Use isometric dot paper to sketch the object. An object is created using eight blocks. It has the following top, front, and side views. Sketch what the object looks like on isometric dot paper. top front side 8 MHR Chapter 5

8 Example 3: Predict and Draw the Top, Front, and Side Views After a Rotation The diagrams show the top, front, and side views of the computer tower. top front side You want to rotate the computer tower 90º clockwise on its base to fit into your new desk. Predict which view you believe will become the front view after the rotation. Then, draw the top, front, and side views after rotating the tower. This diagram shows a 90 clockwise rotation. 90 Solution The original side view will become the new front view after the rotation. top front side You can use a Draw program to create 3-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 about its spine. Then, draw the top, front, and side views after rotating the textbook. 5.1 Views of Three-Dimensional Objects MHR 9

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 a 3-D object. top front side 1. Raina insists that you need to tell her all six views so she can draw your object. Is she correct? Explain why or why not. 2. Are these views correct? Justify your answer. front top side c) For help with #3 and #4, refer to Example 1 on page XXX. 3. Sketch and label the top, front, and side views. a) b) Photo Album 4. Choose the correct top, front, and side view for this object and label each one. A B C D 10 MHR Chapter 5 E F G

10 For help with #5, refer to Example 2 on page xxx. 5. Draw each 3-D object using the views below. a) top front side 8. Choose two 3-D objects from your classroom. Sketch the top, front, and side views for each one. b) top front side 9. Sketch the front, top, and right side views for these solids. a) b) For help with #6 and #7, refer to Example 3 on page xxx. 6. A television set has the following views. c) front front top front side front If you turn the television 90 counterclockwise, how would the three views change? Sketch and label each new view. 7. Choose which object has a front view like this after a rotation of 90º clockwise onto its side. a) set of books b) CD rack 10. Describe two objects that meet this requirement: When you rotate one object 90 degrees, the top, front, and side views are the same as the top, front, and side views of the other object that was not rotated. 11. An injured bumblebee sits at a vertex of a cube with edge length 1 m. The bee moves along the edges of the cube and comes back to the original vertex without visiting any other vertex twice. a) Draw diagrams to show the bumblebee s trip around the cube. b) What is the length, in metres, of the longest trip? MATH LINK Choose one of the essential buildings that you discussed for your new community on page XXX. Draw and label a front, side, and top view. 5.1 Views of Three-Dimensional Objects MHR 11

11 Nets of Three-Dimensional Objects Focus on After this lesson, you will be able to... determine the correct nets for 3-D objects build 3-D objects from nets draw nets for 3-D objects rectangular prism a prism whose bases are congruent rectangles grid paper scissors clear tape rectangular prisms (blocks of wood, cardboard boxes, unit blocks) net a two-dimensional shape that, when folded, encloses a 3-D object Shipping containers help distribute materials all over the world. Items can be shipped by boat, train, or transport truck to any destination using these containers. Shipping containers are right rectangular prisms. Why do you think this shape is used? How do you know if a net can build a right rectangular prism? Here are a variety of possible nets for a right rectangular prism. rectangular prism Literacy Link A right prism has sides that are perpendicular to the bases of the prism. net cube 1. Draw each net on grid paper. 12 MHR Chapter 5

13 Example 2: Build a Three-Dimensional Object From a Given Net Before going to leadership camp, your group needs to put a tent together. Can this net be folded to form the shape of a tent? Strategies Model It Refer to page xxx. triangular prism a prism with two triangular bases each the same size and shape Solution Trace the net onto paper. Cut along the outside edges and fold along the inside edges. Tape the cut edges together to try to build a right triangular prism. The net can be folded to form the shape of a tent. Build a 3-D object using this net. What object does it make? 14 MHR Chapter 5

14 A net is a two-dimensional shape that, when folded, encloses a three-dimensional object. The same 3-D object can be created by folding different nets. net cube You can draw a net for an object by visualizing what it would look like if you cut along the edges and flattened it out. 1. Both of these nets have six faces, like a cube. Will both nets form a cube? Justify your answer. net A net B 2. Patricia is playing the lead role in the school musical this year. She missed Math class while she was performing. She cannot figure out if a net will build the correct 3-D object, and asks you for help after school. 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 puck chocolate bar jewellery box 5.2 Nets of Three-Dimensional Objects MHR 15

15 4. Draw the net for each object. Label the measurements on the net. a) d = 30 mm 7. Match each solid with its net. Copy the nets, then try to create the 3-D objects. 78 mm A ream describes a quantity of approximately 500 sheets of paper. rectangular prism cylinder b) triangular prism A B 28 cm Paper 500 Sheets 21.5 cm 5 cm 5. Draw a net on grid paper for a rectangular prism with the following measurements: length is six units, width is four units, and height is two units. C For help with #6 and #7, refer to Example 2 on page xxx. 6. a) Draw the net on grid paper, as shown. Cut along the outside edges of the net and fold to form a 3-D object. D E b) What is this object called? 8. A box of pens measures 15.5 cm by 7 cm by 2.5 cm. Draw a net for the box on a piece of centimetre grid paper. Then, cut it out and fold it to form the box. 9. You are designing a new mailbox. Draw a net of your creation. Include all measurements. 16 MHR Chapter 5

16 10. Angela designed two nets. 12. The six sides of a cube are each a different colour. Four of the views are shown below. a) Enlarge both nets on grid paper, and build the 3-D objects they form. b) What object does each net form? 11. Hannah and Dakota design a spelling board game. They use letter tiles to create words. Tiles may be stacked (limit of four) on top of letters already used for a word on the board to form a new word. a) Draw a 3-D picture of what these stacked tiles might look like. b) Draw a top view that illustrates the stacked tiles for people reading the instructions. What colour is on the opposite side of each of these faces? a) purple b) blue c) red 13. How many possible nets can create a cube? Sketch all of them. The first one is done for you. MATH LINK When buildings are designed, it is important to consider engineering principles, maximum and minimum height requirements, and budget. a) Create a 3-D sketch of two buildings for your miniature community, one that is a prism and one that is a cylinder. b) Draw a net of each building, including all possible measurements needed to build your miniature. 5.2 Nets of Three-Dimensional Objects MHR 17

17 Surface Area of a Prism Focus on After this lesson, you will be able to... link area to surface area find the surface area of a right prism Most products come in some sort of packaging. You can help conserve energy and natural resources by purchasing products that are made using recycled material use recycled material for packaging do not use any packaging What other ways could you reduce packaging? empty cardboard box (cereal box, granola box, snack box, etc.) scissors ruler scrap paper How 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 Link Exclude The dimensions of an object are measures such as length, width, and height. 18 MHR Chapter 5 3. What are the dimensions and area of each face? overlapping flaps. 4. How can you find the amount of material used to make your cardboard box?

18 Reflect on Your Findings 5. a) How did you use the area of each face to calculate the total amount of material used to make your cardboard box? b) Can you think of a shorter way to find the total area without having to find the area of each face? Explain your method. Example 1: Calculate the Surface Area of a Right Rectangular Prism a) 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? Solution a) 4 cm 10 cm 6 cm 6 cm 10 cm 4 cm surface area the number of square units needed to cover a 3-D object the sum of the areas of all the faces of an object b) The right rectangular prism has faces that are three different sizes. 6 cm front or back 10 cm A = l w A = 10 6 A = 60 The area of the front or back is 60 cm 2. 4 cm top or bottom 10 cm A = l w A = 10 4 A = 40 The area of the top or bottom is 40 cm 2. 4 cm ends 6 cm A = l w A = 6 4 A = 24 The area of each end is 24 cm 2. Area is measured in square units. For example, square centimetres, square metres, etc. c) The total surface area is the sum of the areas of all the faces. The front and back have the same area: A = 60 2 A = 120 The top and bottom have the same area: A = 40 2 A = 80 The two ends have the same area: A = 24 2 A = 48 Total surface area = (area of front and back) + (area of top and bottom) + (area of ends) = = 248 The surface area of the right rectangular prism is 248 cm 2. You could add the areas you calculated first = 124 Each area is the same as the area of one other face, so you could then multiply the total by two = Surface Area of a Prism MHR 19

19 What is the surface area of this right rectangular prism? 16 cm 8 cm 3 cm Example 2: Calculate the Surface Area of a Right Triangular Prism a) Draw the net of this right triangular prism. b) What is the area of each face? c) What is the total surface area? Solution 9 m 2.6 m a) 3 m Strategies Draw a Diagram Refer to page xxx. 3 m 2.6 m 9 m Literacy Link An equilateral triangle has three equal sides and three equal angles. Equal sides are shown on diagrams by placing tick marks on them. b) The bases of the prism are equilateral triangles. The sides of the prism are rectangles. 3 m rectangle triangle 9 m 3 m 2.6 m A = l w A = (b h) 2 A = 9 3 A = (3 2.6) 2 A = 27 A = A = 3.9 The area of one The area of one rectangle is 27 m 2. triangle is 3.9 m MHR Chapter 5

20 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 triangles of the same size. Total surface area = (3 area of rectangle) + (2 area of triangle) = (3 27) + (2 3.9) = = 88.8 The surface area of the right triangular prism is 88.8 m 2. Find the surface area of this triangular prism. 7 cm 9.9 cm 7 cm 2 cm Surface area is the sum of the areas of all the faces of a 3-D object. A1 A6 A2 A3 A4 A5 Surface Area = A1 + A2 + A3 + A4 + A5 + A6, where A1 represents the area of rectangle 1, A2 represents the area of rectangle 2, etc. 1. Write a set of guidelines that you could use to find the surface area of a prism. Share your guidelines with a classmate. 2. A right rectangular prism has six faces. Why do you have to find the area of only three of the faces to be able to find the surface area. Use pictures and words to explain your thinking. 5.3 Surface Area of a Prism MHR 21

21 For help with #3 and #4, refer to Example 1 on page xxx. 3. Find the surface area of this right rectangular prism to the nearest tenth of a square centimetre cm 13.5 cm 5 cm 7. Given the area of each face of a right rectangular prism, what is the surface area? front top side 20 mm 2 12 mm 2 15 mm 2 8. Paco builds a glass greenhouse. 4. Find the surface area of this CD case. 14 cm For help with #5 to #7, refer to Example 2 on page xxx cm 1 cm 5. Calculate the surface area of 2.7 m this ramp in the 1.5 m shape of a right triangular prism. 2.3 m Give your answer to the nearest tenth of a square metre. 0.7 m 1.1 m 1.8 m 0.6 m 2.4 m a) How many glass faces does the greenhouse have? b) How much glass does Paco need to buy? 9. What is the minimum amount of material needed to make the cover of this textbook if there is no overlap? Give your answer to the nearest square millimetre. 10. Jay wants to make a bike ramp. He draws the following sketch. What is the surface area of the ramp? 6. Cheese is sometimes packaged in a triangular box. How much cardboard would you need to cover this piece of cheese if you do not include overlapping? Calculate your answer to the nearest tenth of a square centimetre. 0.9 m 1.6 m 2.2 m 2 m 4.5 cm 3 cm 9 cm 6.4 cm The tick marks on the two sides of the triangle indicate that these sides are equal. 22 MHR Chapter 5

22 11. Dallas wants to paint three cubes. The cubes 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 will Dallas paint if he decides not to paint the bottoms of the three cubes? 12. Tadika has a gift to wrap. Both of these containers will hold her gift. Which container would allow her to use the least amount of wrapping paper? Explain your choice. 5 cm 10 cm 7 cm 30 cm 10 cm 5 cm 13. A square cake pan measures 30 cm on each side and is 5 cm deep. Cody wants to coat the inside of the pan with non-stick oil. If a single can of non-stick oil covers an area of cm 2, how many pans can 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 to hold all ten packages of cards. a) What are the minimum inside dimensions of the container? b) Is there more than one kind of container that would work? Draw diagrams to help explain your answer. 15. a) If the edge length of a cube is doubled, find the ratio of the old surface area to the new surface area. b) What happens if the edge length of a cube is tripled? Is there a pattern? 16. Shelby wants to paint the walls and ceiling of a rectangular room. 2.6 m 6.8 m 4.8 m Type of Paint Size of Paint Can Cost Wall paint 4 L 1 L \$24.95 \$7.99 Ceiling paint 4 L \$32.95 One litre of paint covers 9.5 m 2. a) What is the least amount of paint Shelby can buy to paint the room (subtract 5 m 2 for the door and windows)? b) How much will the paint cost, including the amount of tax charged in your region? MATH LINK For the prism-shaped building you created in the Math Link on page XXX, how much material do you need to cover the exterior walls and the roof of the building? 5.3 Surface Area of a Prism MHR 23

24 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 in your final answer. Pop cans are cylinders. The world s largest Coke can is located in Portage la Prairie, Manitoba. Reflect on Your Findings 5. 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 Cylinder a) Estimate the surface area of the can. b) What is the surface area of the can? Express your answer to the nearest hundredth of a square centimetre? 11 cm Solution 7.5 cm The surface area of the can is found by adding the areas of the two circular 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 There are two circles: 2 48 = 96 The 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 circumference of a circle. 240 The area of the rectangle is approximately 240 cm 2. Estimated surface area = area of two circles + area of rectangle The estimated surface area is 340 cm 2. r 2 means r r The radius of a circle is half the diameter. Literacy Link circle radius centre diameter The formula for the circumference of a circle is C = π d or C = 2 π r. 5.4 Surface Area of a Cylinder MHR 25

25 Strategies Draw a Diagram Refer to page xxx. b) Method 1: Use a Net Draw the net and label the measurements. top side bottom 7.5 cm 11 cm The diameter of the circle is 7.5 cm. Determine the radius = 3.75 The radius of the circle is 3.75 cm. Round your answer at the end of the calculation. Find the area of one circle. A = π r 2 Use 3.14 as an approximate value A for π. A The area of one circle is approximately cm 2. Find the area of two circles = The area of both circles is approximately cm 2. Find the area of the rectangle using the circumference of the circle. A = l w A = (π d) w Replace l with the formula for the circumference of a circle. A A The area of the rectangle is approximately cm 2. Calculate the total surface area. Total surface area = = The total surface area is approximately cm MHR Chapter 5

26 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 ( ) + ( ) S.A S.A The total surface area is cm 2, to the nearest hundredth. This formula incorporates each shape and its area formula to find the surface area. 2 (π r 2 ) + (π d) h two circles circle area rectangle area formula formula (length is the circumference of a circle; width is the height of the cylinder) Calculate the surface area of this cylinder to the nearest tenth of a square centimetre. 9 cm 55 cm Literacy Link The abbreviation S.A. is often used as a short form for surface area. Example 2: Use the Surface Area of a Cylinder Calculate the surface area of this totem pole, including the two circular bases. The pole stands 2.4 m tall and has a diameter of 0.75 m. Give your answer to the nearest hundredth of a square metre. Solution The cylinder has two circular bases. The area of one circle is: A = π r 2 A r = d 2 A The area of the circle is approximately m 2. There are two circles, so the area of both circles is approximately m 2. Calculate the total surface area. S.A S.A The total surface area is approximately 6.54 m 2. The side of the cylinder is a rectangle. The area of the rectangle is: A = (π d) h A A The area of the rectangle is approximately m 2. Replace one dimension with the formula for the circumference of a circle. Calculate the surface area of a cylindrical waste backet without a lid that measures 28 cm high and 18 cm in diameter. Give your answer to the nearest square centimetre. This metal totem pole was created by Todd Baker, Squamish Nation. It represents the Birth of the Bear Clan, with the princess of the clan on the top half and the bear on the bottom half. 5.4 Surface Area of a Cylinder MHR 27

27 The surface area of a cylinder is the sum of the areas of its faces. A net of a cylinder is made up of one rectangle and two circles. To find one of the dimensions of the rectangle, calculate the circumference of the circle. The length of this side is the circumference of the circle C = π d or C = 2 π r 1. What are the similarities and differences between finding the surface 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 find the surface area of a cylinder. For help with #3 to #7, refer to Examples 1 and 2 on pages xxx xxx. 3. a) Draw a net for this cylinder. b) Sketch a different net for this cylinder. 4. Estimate the surface area of each cylinder. Then, calculate each surface area to the nearest tenth of a square centimetre. a) d = 7 cm 30 cm b) r = 10 cm 22 cm 5. Find the surface area of each object to the nearest tenth of a square unit. a) b) d = 2.5 cm 16 cm wooden rod flag pole d = m 16 m 6. Use the formula S.A. = 2 (π r 2 ) + (π d h) to calculate the surface area of each object. Give each answer to the nearest hundredth of a square unit. a) d = 2.5 cm b) d = 5 cm 10 cm You can simplify the formula: S.A. =2 (π r 2 ) + (π d h) = 2πr 2 + πdh 7 cm 28 MHR Chapter 5

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