Perimeter, Area, and Volume. Chapter. Big Idea. Learning Goals. Essential Question. Important Words

Transcription

1 SHAPE and space Perimeter, Area, and Volume Chapter Big Idea 10 Understanding perimeter, area, and volume helps me to describe the world and solve real-world problems. Learning Goals I can develop a formula for determining the perimeter of polygons. I can apply a formula for determining the perimeter of polygons. I can develop a formula for determining the area of rectangles. I can apply a formula for determining the area of rectangles. I can develop a formula for determining the volume of right rectangular prisms. I can apply a formula for determining the volume of right rectangular prisms. Essential Question How can I use perimeter, area, and volume to describe the world around me and solve problems? Important Words area formula perimeter volume

2 Perimeter Study Draw, build, measure, describe, and calculate the perimeter of different polygons. Example: Find the perimeter of the following polygon. Sandeep s strategy: I measured each side: 1 cm, 2 cm, 2 cm, 2 cm. I added all the sides together to get the perimeter: = 7 The perimeter of this trapezoid is 7 cm. Mariam s strategy: I measured each side: 1 cm, 2 cm, 2 cm, 2 cm. I saw that three of the sides are 2 cm. I wrote an equation to multiply the 2 cm by the three sides like this: P = (3 x 2 cm) + 1 cm P = 6 cm + 1 cm P = 7 cm The perimeter of this trapezoid is 7 cm. 11. Four polygons are shown below. i. ii. iii. iv. Perimeter is the distance around a shape. a. Estimate to list the polygons from shortest perimeter to longest perimeter. b. Measure the sides and calculate the perimeter of each polygon. 224 CHAPTER 10: Perimeter, Area, and Volume

3 Perimeter Study (continued) 12. Write a formula that could be used to calculate the perimeter of each polygon. A formula is an equation that uses variables to express a rule. a. b. c. d. 13. Draw a polygon that could match each equation. a. 3 cm + 2 cm + 4 cm = P b. P = 2(3 cm) + 2 cm c. (3 2 cm) + (3 3 cm) = P d. P = 1 cm + 3 cm + 1 cm + 2 cm + 2 cm + 1 cm e. P = 2 (2 cm + 4 cm) f. 3 cm + 3 cm + 3 cm + 3 cm = P 14. Use a geoboard to build a rectangle that has a perimeter of 14 units. a. Record the possible dimensions of the rectangle. b. Use a formula to find the perimeter of each rectangle. 15. Match each shape to a formula that could be used to calculate the perimeter. a. P = a + b + c b. P = 2b + 2h c. P = 6s d. P = a + b + c + d i. ii. iii. iv. 16. Gavin thinks the equation P = 3 4 could be used to find the perimeter for two different shapes. What shapes might this equation describe? CHAPTER 10: Perimeter, Area, and Volume 225

4 Perimeter Study (continued) 17. Explain whether P = 4s could be used to find the perimeter for each of the quadrilaterals below. a. b. c. d Solve each perimeter riddle a. A polygon has six sides. Two of the sides are the same length. The other four sides are all the same length. The four equal sides are double the length of the two equal sides. The short sides measure 2 centimetres in length. What is the perimeter of the polygon? b. A polygon has four sides. All four sides are the same length. The total perimeter is 16 centimetres. How long is each side? 19. A polygon has three sides. All three sides are different lengths. The shortest side measures 2 centimetres and the longest side measures 5 centimetres. What could the perimeter be? Use dot paper or grid paper to draw all the possible quadrilaterals that have a perimeter of 12 centimetres and that have side lengths that are whole numbers. 11. Write a formula that could be used to find the perimeter of each type of quadrilateral in question Write a general rule for calculating the perimeter of: a. an equilateral triangle. b. a square. c. a regular pentagon. d. a regular octagon. 13. How can formulas help you calculate the perimeter of polygons? I can develop a formula for determining the perimeter of polygons. I can apply a formula for determining the perimeter of polygons. 226 CHAPTER 10: Perimeter, Area, and Volume

5 Perimeter Problems Draw, build, measure, describe, and calculate perimeter to solve problems from real-life contexts, such as gardening, building, planning, and sewing. Example: Sandra s family is building a new flower bed in their garden. They bought 12 metres of wood to outline the perimeter. They have decided they do not want a rectangular flower bed. What could their flower bed look like? Sandra s strategy: I am going to make the garden a regular hexagon. A hexagon has six sides. 12m 6 = 2m Each side could be 2 metres. The garden would look like this: Sandra s dad s strategy: I am going to make the garden a quadrilateral, but not a rectangle. I found four numbers that add up to 12. 4m + 1m + 5m + 2m = 12m Gerome is planning a vegetable garden for the corner of his backyard as shown below: 4 m 8.9 m 8 m Gerome wants to plant flowers around the edges of his garden. What is the perimeter of the garden? CHAPTER 10: Perimeter, Area, and Volume 227

6 Perimeter Problems (continued) 12. This is the plan for Martin s new deck: 3 m 0.6 m 1 m Stair Stair 1.2 m 1.8 m 0.8 m Fire Pit 0.3 m 2 m 0.6 m a. Find the perimeter of the fire pit. b. What formula could you use to calculate the perimeter of the fire pit? c. Explain whether you could use the same formula to find the perimeter of the deck. d. What is the perimeter of the entire deck? 13. Elmer s garden is a rectangle that is 10.1 metres long and 4.2 metres wide. To keep rabbits out of his garden, Elmer built a rectangular fence that is 11.2 metres by 5 metres. How much greater is the perimeter of the fence than the perimeter of the garden? 14. Pierre has just moved into a new house and is trying to arrange the furniture in his room. His room is a rectangle that has a perimeter of 12 metres. How many metres long might each wall be? 15. The shower in Katherine s bathroom is shown below: Katherine knows the perimeter of the shower measures 7.4 metres. How long is the fifth side? 1 m 2 m 1 m 2 m 228 CHAPTER 10: Perimeter, Area, and Volume

7 Perimeter Problems (continued) 16. Marie wants to sew ribbon around a rectangular blanket that she knows has a perimeter of 900 centimetres. She measured one edge to be 175 centimetres. How long are the other sides of the blanket? 17. Kendra is a wedding planner. She is trying to decide how to arrange the tables around the dance floor. The dance floor is a rectangle that measures 12 metres by 18 metres. Each table is three metres long. How many tables can she fit around the dance floor? 18. The Samulla family is renovating their backyard. Their backyard is 30 metres long and 20 metres wide. Fence panels are 2 metres long. How many fence panels will the Samulla family need to enclose the entire backyard? 19. Matteous is making a quilt for his bed. His bed is 1.5 metres wide and 2 metres long. He wants the quilt to hang down 25 centimetres on all four sides of the bed. What should the perimeter of his quilt be? 10. Imagine you lined up all the people in your class around the perimeter of your classroom. Choose to answer one of the following questions. a. How much wall space would each person get? b. How much space would there be between each person? 11. Henry drew a square on centimetre grid paper that had sides 2 centimetres long. Nevaeh wanted to draw a square with twice the perimeter of Henry s square. Should she double the length of each side? Why? 12. Is there one perimeter formula that works for all shapes? Explain. I can develop a formula for determining the perimeter of polygons. I can apply a formula for determining the perimeter of polygons. CHAPTER 10: Perimeter, Area, and Volume 229

8 Area Formulas Draw, build, measure, describe, and calculate area of rectangles. Example: Find the area of a 5 centimetre by 12 centimetre rectangle. Lou s strategy: I drew the rectangle on centimetre grid paper. I have five rows of 12 square centimetres each. 5 x 12 cm 2 = 60 cm 2. The area is 60 square centimetres. Damien s strategy: I built a rectangle out of interlocking cubes. I made 12 rows of five cubes each. 12 x 5 = (10 + 2) x 5 = 10 x x 5 = = 60 I know the area of this rectangle is 60 square centimetres. 230 CHAPTER 10: Perimeter, Area, and Volume

9 Area Formulas (continued) Siobahn s strategy: I know the formula A = l x w can be used to find the area of a rectangle if you know the lengths of two perpendicular sides of the rectangle. A = l x w A = 5 cm x 12 cm A = 60 cm On dot paper or grid paper, draw a rectangle that has the following measurements. Calculate the area of each rectangle. a. length = 4 cm, width = 6 cm b. base is 9 cm long, height is 2 cm c. rectangle is 3 cm long and 3 cm tall d. length = 6.5 cm, width = 4 cm Area is the flat space covered by a shape. 12. Calculate the area of each rectangle shown below. a. b. c. d. CHAPTER 10: Perimeter, Area, and Volume 231

10 Area Formulas (continued) 13. Study the rectangles shown below. Copy and complete the chart, then calculate the area of each rectangle. a. b. c. d. Rectangle Length Width Area a. b. c. d. 14. Jamie knew the perimeter formula of some rectangles and used them to help her calculate their areas. Draw each rectangle and check Jamie s work. If it is not correct, explain her mistake. a. P = 2(7 + 2) A = 7 2 = 14 cm 2 b. P = A = 3 3 = 9 cm 2 c. P = 2(1) + 2(4) A = 1 4 = 4 cm 2 d. P = A = = 160 cm CHAPTER 10: Perimeter, Area, and Volume

11 Area Formulas (continued) 15. Tyrell drew the rectangles below and arranged some square centimetre tiles in each rectangle, but he did not have enough tiles to fill each rectangle. Calculate the area of each rectangle Tyrell drew. a. b. c. d. 16. Find the missing information in the table below. Length Width Area 4 cm 5 cm a 6 cm b 12 cm 2 c 7 m 2.8 m If you double the length of each side of a square, how does the area change? Build a model or draw a picture to support your answer. CHAPTER 10: Perimeter, Area, and Volume 233

12 Area Formulas (continued) 18. Estimate to order the rectangles below from one having the smallest area to one having the largest area. a. b. c. d. 19. Calculate the area of each rectangle in question 8 to check your order. 10. Violet found the area of the two blue rectangles below. How could she use them to find the area of the yellow polygon? 20 cm 20 cm 12 cm 240 cm 2 20 cm 12 cm 10 cm 8 cm 80 cm 2 10 cm 11. In what other ways could Violet have found the area in question 10? 12. How are the perimeter formula and the area formula similar? different? I can develop a formula for determining the area of rectangles. I can apply a formula for determining the area of rectangles. 234 CHAPTER 10: Perimeter, Area, and Volume

13 Area Problems Draw, build, measure, describe, and calculate the area of rectangles to solve problems from real-life contexts, such as laying flooring, painting, planning a garden, and baking. Example: Maxine is planning to paint her bedroom. She wants to paint the largest wall of her room a bright colour. The wall is four metres long and three metres tall. a. How much area will the paint have to cover? I used grid paper to draw a model of the wall of her bedroom. Then I used the formula A = l x w. A = 3 m x 4 m A = 12 m 2 b. One can of paint covers 10 square metres. How many cans of paint will she need to buy? One can of paint covers 10 m 2. Two cans of paint will cover = 20 m 2. Maxine needs to buy two cans of paint, but she will have lots of paint left over. 11. Orenda wants to build a giant puzzle which will cover 1000 square centimetres when it is finished. Her table measures 61 centimetres by 122 centimetres. Is it likely the puzzle will fit on the table? How do you know? 12. George is designing a flower bed as shown. 3 m a. How much area will the flower bed cover in total? b. What is the perimeter of the flower bed? c. How might George use each of these measures? 1.2 m CHAPTER 10: Perimeter, Area, and Volume 235

14 Area Problems (continued) 13. The Samulla family is putting down sod in their backyard. Their backyard is 30 metres long and 20 metres wide. Each roll of sod covers 15 square metres. a. What do you need to know to determine the number of rolls of sod the Samulla family will have to order, to cover the entire backyard? b. How many rolls of sod should they order? c. Will they have any sod left over? If so, how much? 14. Jacquie is retiling her bathroom backsplash. She has blue, yellow, and white tiles. The length of the backsplash is 175 centimetres and the height is 35 centimetres. a. What is the area of the backsplash? b. The tiles are each 5 cm 5 cm squares. What is the area of one tile? c. She has 100 white tiles, 75 blue tiles, and 75 yellow tiles. Will she have enough tiles to cover the backsplash? 15. Beth is making brownies. Her baking tray measures 35 centimetres by 65 centimetres. a. How many brownies can she cut from one tray if each one measures 10 square centimetres? b. If Beth made mini-brownies that only had an area of 5 square centimetres each, how many would she be able to cut from one tray? c. Which size of brownie should Beth cut? Explain. 16. Your class is planning to bake coconut macaroons. When the cookies come out of the oven, they need to be placed on the work table to cool. You want to cover the work table with wax paper to keep the cookies from sticking. Will this roll of wax paper cover your table? 65 cm 185 cm 32 cm 3 m 236 CHAPTER 10: Perimeter, Area, and Volume

15 Area Problems (continued) 17. Katherine is building a new bathroom. She has already bought the fixtures. The bathtub is 1 metre by 2 metres, the toilet needs a space 1 metre by 1 metre, the vanity is 1 metre by 2 metres, and the shower is 1 metre by 1.5 metres. a. Design what Katherine s bathroom might look like. b. What is the area of the room you designed? 18. Makayla is making a pair of curtains for her bedroom. The window is 1 metre tall and 3 metres wide. Each curtain is twice as tall as the window and one-and-a-half times as wide as the window. How much fabric will she need for each curtain? 19. Argen wants to build a deck like the one shown below. To order the supplies, he needs to know both the perimeter and the area of the deck. How could Argen find these two measures? 8 m 5 m 3 m 2 m 10. What does the 2 in cm 2 mean? 11. How are perimeter and area different? I can develop a formula for determining the area of rectangles. I can apply a formula for determining the area of rectangles. CHAPTER 10: Perimeter, Area, and Volume 237

16 Volume Draw, build, measure, describe, and calculate the volume of rectangular prisms. Example: Calculate the volume of this rectangular prism. 2 cm 4 cm David s strategy: I used centimetre cubes to build a prism that had the same dimensions. Then I counted the cubes in one layer. The base layer had four rows of three cubes each, which equals 12 cubes or 12 cubic centimetres. My prism had two layers, so in total that was 24 cubes. Each cube is 1 cubic centimetre, so the volume of the prism was 24 cubic centimetres. 3 cm Kendra s strategy: I calculated the area of the base. A = l x w A = 4 cm x 3 cm A = 12 cm 2 I then multiplied the area of the base by the height. V = (base area) x h V = 12 cm 2 x 2 cm V = 24 cm 3 The vol ume of the prism was 24 cubic centimetres. Olive s strategy: I used the formula V = l x w x h. V = 4 cm x 3 cm x 2 cm V = 24 cm 3 The volume of the prism was 24 cubic centimetres. 238 CHAPTER 10: Perimeter, Area, and Volume

17 Volume (continued) 11. Each prism below is built with centimetre cubes. Find the volume of each prism. The volume of an object is the amount of space the object takes up. a. b. c. 12. Find the volume of each prism shown below. a. b. 4 cm 6 cm 2 cm 8 cm c. 5 cm 7 cm 1 cm 12 cm 6 cm 13. Compare the strategies you used in question 1 with those you used in question Build a model or draw a picture to show at least two different rectangular prisms for each volume. a. V = 36 cm 3 b. V = 48 cm What strategies did you use to answer question 4? 16. Compare the prisms you created in question 4. CHAPTER 10: Perimeter, Area, and Volume 239

18 Volume (continued) 17. Calculate the volume of each rectangular prism. a. b. Base area = 45 cm 2 12 cm 12 cm 9 cm 5 cm 18. What did you notice about the volumes of the prisms in question 6? Explain your observations. 19. Noah built the prism shown using centimetre cubes. How can he find the number of cubes in the prism without taking it apart? 10. How does the strategy you described in question 9 relate to the volume formula you used? 11. Use a formula to find the volume of the prisms below. a. b. 2.5 cm 4 cm 2 cm 1.5 cm 1 cm 5 cm 12. Why might using a formula be an efficient strategy for calculating the volume of each prism in question 11? 240 CHAPTER 10: Perimeter, Area, and Volume

19 Volume (continued) 13. Zayden built a prism with a volume of 432 cubic centimetres, as shown. What is the height of the prism? 6 cm 12 cm 14. Adam found the volume of the rectangular prisms below, but wants someone to check his work. Explain whether Adam is correct. a. V = l w h V = 5 cm 4 cm 10 cm V = 200 cm 3 b. V = (area of the base) height V = (12 cm 2 cm) 2 cm V = 24 cm 2 2 cm V = 48 cm Use what you know about volume to complete the chart below. Length Width Height Volume a 6 7 b c CHAPTER 10: Perimeter, Area, and Volume 241

20 Volume (continued) 16. Explain how you could use the volumes of the two green prisms below to find the volume of the third prism. V = cm 3 V = cm 3 45 cm 40 cm 60 cm 90 cm 30 cm 70 cm 60 cm 40 cm 90 cm 45 cm 70 cm 60 cm 17. How are the two formulas for volume in question 14 related? I can develop a formula for determining the volume of right rectangular prisms. I can apply a formula for determining the volume of right rectangular prisms. 242 CHAPTER 10: Perimeter, Area, and Volume

21 Volume Problems Draw, build, measure, describe, and calculate the volume of rectangular prisms to solve problems from real-life contexts, such as cooking, organizing, or gardening. Example: Jared is sending shoes to children overseas as part of a service project. The shoes have a total volume of 5000 cubic centimetres. The box he has to send them in measures 15 cm by 30 cm by 12 cm. Will the shoes fit inside the box? First I drew a picture of the box and found the area of the base. A = 15 cm x 30 cm A = 450 cm 2 V = 450 cm 2 x 12 cm V = 5400 cm 3 30 cm 15 cm I checked my work by using another formula for volume. V = l x w x h V = 15 cm x 30 cm x 12 cm V = 5400 cm 3 12 cm The volume of the box is greater than the volume of the shoes, so the shoes should fit inside the box if they are packed carefully. 11. Olivia is using plastic containers to organize her bedroom. The dimensions of each container are 15 cm x 32 cm x 14 cm. What is the volume of one plastic container? 12. Liam found two boxes in his basement. One box measures 2 metres by 1 metre by 1.5 metres and the other measures 9 centimetres by 5 centimetres by 4 centimetres. a. Find the volume of each box. b. Describe what each box might be used for. CHAPTER 10: Perimeter, Area, and Volume 243

22 Volume Problems (continued) 13. Madison s family just bought a hot tub. The instructions say they need to fill it with 18 cubic metres of water. What might the dimensions of the hot tub be? 14. Anthony is making lemonade for a backyard party. The lemonade jug is 30 centimetres tall and the base of the jug has an area of 260 square centimetres. a. How much lemonade can he make at one time? b. After serving his friends, half of the lemonade is gone. What volume of lemonade is left? 15. Isabella is making a cake for her Mom s birthday. She is not sure whether she should use the small pan or the large pan. The recipe says it makes a volume of 6160 ml, but Isabella knows that 1 ml = 1 cm 3, so the volume is 6160 cubic centimetres. The small cake pan measures 22 cm 22 cm 8 cm. The large cake pan measures 35 cm 35 cm 10 cm. Explain which pan you think IsabeIla should use. 16. What strategies could Anya use to find the volume of the prism shown below? 30 cm 20 cm 10 cm 20 cm 10 cm 10 cm 17. Ethan is bringing Jello blocks to his class party. To make Jello blocks he makes Jello in a baking tray then cuts it into rectangular prisms. One large package makes about 700 cubic centimetres of Jello. Ethan s baking tray measures 28 centimetres by 18 centimetres by 4 centimetres. a. Will one package of Jello be enough to fill the tray? b. Ethan wants each Jello block to be a 4 cm 4 cm 4 cm cube. How many Jello blocks will he be able to make from one tray? c. Will Ethan have enough Jello blocks for each of the 26 students in his class to have one block? 244 CHAPTER 10: Perimeter, Area, and Volume

23 Volume Problems (continued) 18. Memphis has a wading pool in his backyard. The area of the base of the wading pool is 8000 square centimetres. How deep will the water be when he fills the wading pool with cubic centimetres of water? 19. Bilal built a prism using interlocking cubes as shown below. The edge of each interlocking cube measures 2 centimetres. What is the volume of the prism? 10. Sophia built a raised flower bed for her favourite flowers. She used bricks to build a wall along one side of the flower bed. Each brick measures 10 centimetres long, 20 centimetres wide and 6 centimetres tall. a. What is the volume of each brick? b. She made the wall six bricks tall and 30 bricks long. What is the volume of the wall? CHAPTER 10: Perimeter, Area, and Volume 245

24 Volume Problems (continued) 11. Alice is building a planter, as shown below, for her yard. 0.2 m 1 m 2 m a. Alice needs to know the perimeter of the planter to order the wood for building the sides. What is the distance around the planter box? b. Before she fills the planter box with flowers, she needs to cover the bottom of the box with landscaping fabric. How much fabric does she need to buy? c. The soil to fill the planter costs $52/cubic metre. How much will Alice need to spend on soil? 12. What does the 3 mean in cm 3? I can develop a formula for determining the volume of right rectangular prisms. I can apply a formula for determining the volume of right rectangular prisms. 246 CHAPTER 10: Perimeter, Area, and Volume