Chapter 3. Length, Area, and Volume

Transcription

1 1 Chapter 3 Length, Area, and Volume 3.1 Systems of Measurement 3.2 Converting Measurements 3.3 Surface Area 3.4 Volume 34 Name:

2 2

3 3 3.1 Systems of Measurement In Canada we use systems of measurement: The SI (Système International d unités) which is the and we the. We usually use the SI system (metric system) in our daily lives but the imperial system is used in most trades. For example, plumbers and builders take measurements in. Therefore, it is important to understand both systems. In the metric system the base unit for measuring length is the (m). The base unit for (l). The metric system is based on multiples of. This is important because for any SI unit, it can be converted to another SI unit by either by 10. Si Units Prefixes: : means one hundredth So 1 centimetre of a meter. Kilo: So a kilometre means meters.

4 4 : Is the distance around an object or room, add up all the sides. For a square: all sides are. 5 m 5 m 5 m 5 m For a rectangle: where is the length and the is the width. 8 m 3 m 3 m 8 m Example 1: What is the perimeter of this figure?

5 5 Circular Perimeter or Circumference: The circumference is just a special word for perimeter (distance) around a circular object. To calculate circumference you need to know a few things about a circle. The diameter of the circle is the distance from one side to the other. The radius is the distance to an outside edge. r = d = The symbol: Note: we will use the π button on your calculator. Circumference (distance (perimeter) around a circle): Example 1: Find the perimeter. 2m or 4m Example 2: The sides of the flower garden show below are 4m long. Each end is a semi-circle with a diameter of 2m. What is the perimeter of the flower garden? Worksheet: Perimeter and Circumference

6 6 Perimeter and Circumference Worksheet 1. Calculate the perimeters of the following diagrams. a) b) c) 2. Darma is edging a tablecloth with lace. The tablecloth is 210 cm by 180 cm. How much lace does she need?

7 7 3. Garry installs a wire fence around a rectangular pasture. The pasture measures 15 m by 25 m, and he uses three rows of barbed wire. How much wire did he use? 4. Chantal is building a fence around her swimming pool. The pool is 25 ft long and 12 ft wide, and she wants a 6-ft wide rectangular walkway around the entire pool. How much fencing will she need? 5. What is the circumference of a circular fountain if its radius is 5.3 m? 6. Johnny wants to put Christmas lights along the edge and peak of his roof. How many meters of lights will he need? 7. Harrison uses coloured wire to make a model of the Olympic symbol (5 interlocking circles). If each circle has a radius of 35 cm, how much wire does he need for the rings? Answers: 1a) 53.6 cm b) 43.6 cm c) 7.6 m 2) 780 cm 3) 240 m 4) 122 ft 5) 33.3 m 6) 104 m 7) cm

8 8 Working With Systems of Measurement: The Imperial System Imperial System: (American System) which we also use in the trades. The base unit for length is a. And the base unit for volume is a. The imperial system is a decimal system. Instead each group of units has a particular relationship. Adding Imperial Units: Unit Inch Foot Yard Mile Some common Imperial Units: Length Abbreviation When we add lengths that contain more than 1 unit we need to convert the length into 1 unit. Example 1: I want to find out how much pipe I need to buy. If I need a piece 2 long, a piece 5 4 long and 7 2 long. How much pipe do I need? It s really important to remember that there are 12 inches in a foot. Solution: Add. Add Convert the inches to feet by dividing by 12. And the remainder is the left over inches. Let s say you needed 6 4 in and 2 10 and 3 5 of pipe. What is the total amount of feet and inches?

9 9 Example 2: Maria, a seamstress, is sewing bridesmaids dresses. She orders the fabric from the United States, where fabric is measured in yards. Each dress requires 3¾ yards of silk, 1½ yards of lace fabric, and 7¼ yards of trim. How much of each type of material does Maria need to make 5 dresses? Example 3: Lesley is trying to calculate how much baseboard she will need for the room shown below. What is the minimum amount of baseboards she will need? Worksheet: Working with Systems of Measurement

10 10 Working with Systems of Measurement Worksheet 1. Barry is buying some lumber to finish a project. He needs 3 pieces of 2 by 4 that are each 4½ feet long, and 10 pieces of 2 by 2 that are each 5¼ feet long. How much of each does he need in total? 2. Benjamin is replacing some plumbing pipes. He needs 3 pieces of copper pipe: one piece is 2 feet long, one is 5 feet 7 inches long, and one is 4 feet long. How much copper pipe does he need if he loses 1 inch when he cuts the pipe and he can only buy it in even numbers of feet? 3. If each board in a fence is 6 inches wide, how many of them will James need to fence a playground that is 60 feet wide by 125 feet long? 4. A pet shop stores 5 pet cages that are 2 8 wide, 3 cages that are 4 6 wide, and 2 cages that are 1 8 wide. Can these cages fit side by side along a wall that is 30 long? How much room will be left over or by how much do the cages go over 30? 5. A circular garden is 6 4 in diameter. To plant a geranium approximately every foot along the circumference, how many geraniums are needed?

11 11 6. The height of a basement ceiling is 7 2. A 6 deep heating pipe runs across the middle. To enclose it, there must be a 1 inch space between the pipe and the drywall. Will Craig, who is 6 6 tall, be able to walk under the finished pipes? By how much will he be short or over? 7. You are building a fence around your vegetable garden in your backyard. If the garden is 12 8 long and 4 6 wide, what is the total length of fencing you will need? 8. Marjorie is building a dog run that is 25 8 long and 8 8 wide. How much fencing will she need if the opening is 3 6 wide and will not need fencing? 9. A package of paper is 2 high and 8.5 wide. If a warehouse shelf is 1 5 high and 6 long, how many packages of paper can be put on the shelf? Answers: 1) 13½ of 2 by 4 and 52½ of 2 by 2 lumber 2) 12 ft 3) 740 boards 4) No 2 too long 5) 20 geramiums 6) Yes by 1 in 7) ) ) 64 packages 3.1 Quiz

12 Converting Measurements We need to be able to convert metric measurements to larger or smaller metric measurements. In addition, since we live close to the United States we need to know how to convert units from imperial units to larger or smaller imperial units. Finally, we need to convert metric to imperial and vice versa. A. Converting Metric Units: Recall that the most common metric units of length used are the kilometre (km), the metre (m), the centimetre (cm) and the millimetre (mm). These units of length are related as follows: 10 mm = 1 cm 100 cm = 1 m 1000 m = 1 km To convert length from a larger unit into a smaller unit, multiply by the relevant power of 10. To convert length from a smaller unit into a larger unit, divide by the relevant power of 10. Examples: Convert the following measurements to the indicated units: a) 7 cm to mm b) 8 m to cm c) 9000 m to km d) 2.5 m to mm Worksheet: Converting Metric Units

13 13 Converting Metric Units Worksheet Convert the following measurements to the units indicated: a) 25 m to cm g) 8.26 m to mm b) 6 m to mm h) 4.75 km to mm c) 4 km to cm i) 975 m to km d) 3 km to mm j) 650 cm to m e) 62.8 cm to mm k) 8000 mm to m f) 48.7 m to cm l) cm to km Answers: a) 2500 cm b) 6000 mm c) cm d) mm e) 628 mm f) 4870 cm g) 8260 mm h) mm i) km j) 6.5 m k) 8 m l) 9.5 km

14 14 B. Converting Imperial Units To convert imperial units, you must use a conversion factor. A unit conversion factor is a fraction that is equal to 1. The numerator (top) of the fraction contains the units of the unit you want to convert to. Table of Conversions: 12 inches = 1 foot 36 inches = 1 yard 5280 feet = 1 mile 1760 yards = 1 mile 1 inch = 2.54 cm 1 foot = 30.5 cm 1 foot = 0.305m 1 yard = 3 feet 1 yard = 0.915m 1 mile = 1.6km Whenever we do conversions it s important to put whatever we want we put on top and whatever we have we put on the bottom. Example 1: Convert 116 inches to feet. Example 2: Convert 25 miles to yards.

15 15 Practice: 12.5 inches x. 76 cm x. 7 yards x 149 inches x 20 ft x Worksheet: Converting Imperial Units Converting Imperial Units Worksheet Convert the following measurements to the indicated units to 2 decimal places. a) 5 cm to inches b) 15 m to feet

16 16 c) 10 m to yards i) 3 feet to centimetres d) 10 km to miles j) 2 yards to meters e) 20 feet to meters k) feet to miles f) 25 yards to meters l) 5 feet 8 inches to inches g) 10 feet to meters m) 2000 yards to miles h) 100 km to miles n) 200 miles to km Answers: a) 1.97 in b) ft c) yd d) 6.22 mi e) 6.10 m f) m g) 3.05 m h) mi i) cm j) m k) 5.68 mi l) 68 in m) 1.14 mi n) km

17 17 C. Converting Units Word Problems Many everyday situations require conversions between units. Trades, industry, and many businesses will convert units on a daily basis. Examples: 1) Mary is delivering a load of goods from Vancouver BC to Seattle WA, and then in Seattle she is picking up another load to deliver to Albuquerque, NM. The distance from Vancouver to Seattle is 220 km and the distance from Seattle to Albuquerque is 1456 mi. The odometer in Mary s truck records distance in kilometers. a) What is the total distance she will travel in kilometers? b) If her odometer read km when she left Seattle, what did it read when she left Vancouver? c) What will her odometer read when she reaches Albuquerque?

18 18 2) Mark owns a carpet store and sells hallway runners for $9.52/linear foot. a) How much is this per linear yard? b) How much is the per linear metre? c) Ralph needs 3.9 m of the runner for his hallway. How much will it cost? Converting Units Word Problem Worksheet Converting Units Word Problem Worksheet 1) Suzanne purchased tiles for her patio that are 8 by 4. She measured her patio in metres and wants to convert the tile dimensions to SI units. What are the dimensions of the tiles in centimeters? 2) Ben owns an older American truck. The odometer shows distance travelled in miles. On a recent trip to deliver produce for his employer, he drove 1564 miles. His employer pays him $0.89/km for the use of his own truck. How much will he be reimbursed for the use of his truck for the trip?

19 19 3) A school custodian must mark off a field that is 150 ft by 85 ft. His tape measure is marked in metres. What are the dimensions of the field in metres (to the nearest tenth of a metre)? 4) Jeff knows that his semi-trailer truck is 3.3 m high. A tunnel is marked as Max height: Will Jeff s truck fit through the tunnel? 5) Carla needs 3.5 m of cloth. If the cloth she wants to purchase costs $9.78/yd, how much will the cloth cost? 6) John is a picture framer. He is framing a picture that is 24 inches by 32 inches with a frame that is 2.5 inches wide. What is the outer perimeter of the framed picture: a) In inches? b) In feet and inches? c) In yards, feet, and inches? Answers: 1) 20.3 cm by 10.2 cm 2) $ ) 45.7 m by 25.9 m 4) no too tall 5) $ ) a) 132 in b) 11 ft c) 3 yd 2 ft

20 20 Converting Units Review Worksheet 1. Convert the following imperial measurements: a) 42 inches to feet d) 5 miles to yards b) 84 inches to yards and feet e) 6 feet to inches c) 96 inches to yards f) 1.2 yards to inches 2. Convert the following metric measurements. Recall: a) 5000 metres to kilometres c) 65 centimetres to metres b) 25 centimetres to millimetres d) 150 millimetres to centimetres

21 21 3. Convert the following measurements. a) 16 feet to metres d) 220 yard to meters b) 120 metres to yards e) 45 miles to kilometers c) 26 inches to centimeters f) 150 centimeters to feet 4. Charlie drove from Calgary to Saskatoon, which is a distance of 620 km. How far is this in miles? Answers: 1) a) 3.5 ft b) 2 yd 1 ft c) 2.67 yd d ) 8800 yd e) 72 in f) 43.2 in 2) a) 5 km b) 250 mm c) 0.65 m d) 15 cm 3) a) 4.88 m b) yd c) cm d) m e) km f) 4.92 ft 4) miles 3.2 Quiz

22 Surface Area A. Introduction to Formula: Surface Area of 2D Geometric Figures Formulas are designed so that we can find a measurement of some sort by using a combination of other numbers or constants. Area is a measure of there is on a flat surface. Examples: 1. Calculate the perimeter and area of the following figures: (a) (b)

23 23 2. Calculate the circumference and area of the circles. (a) (b) 3. Sumo is a traditional Japanese sport. The area of a circular sumo ring is m 2. What is the radius of the ring? Worksheet: Working with Formulas

24 24 Working with Formulas Worksheet 1) Determine the perimeter and area of the following geometric figures. (a) (c) (b) (d) 2) Calculate the circumference and area of the following circles. (a) (b) Answers: 1) a) 73.4 ft; 336 ft 2 b) 7.2 mi; 2.16 mi 2 c) 25.2 ft; ft 2 d) 8.7 mi; 2.63 mi 2 2) a) 62.2 cm; cm 2 b) in; in 2

25 25 B. Surface Area of 3D Geometric Figures Rectangular Prism Cylinder +

26 26 Sphere or. Cone 10 ft. Square-Based Pyramid Worksheet: Surface Area

27 27 Surface Area Worksheet 1) Determine the surface area of the following: (a) (b) (c) (d)

28 28 (e) (f) (g) (h) (i) A square pyramid measuring 9 yd along the base with a slant height of 12.8 yd. Answers: 1) a) 90 in 2 b) m 2 c) cm 2 d)442.3 mi 2 e) 50.3 m 2 f) cm 2 g) in 2 h) ft 2 i) yd Quiz

29 Volume A. Calculating Volume The of a solid is a measure of how much it occupies. The is measured in (i.e. and ). In the metric system the base unit for volume is the. In the imperial system the base use for volume is a pint, but we also use cubic inches, feet and yards. What does the word cubic mean? When we measure a object it has. The volume of any object can be determined by the formula: For a right rectangular prism: Example 1: Find the volume of a fish tank that has a length of 70 cm, width of 45 cm, and a height of 60 cm. Example 2: A gas tank (a rectangular prism) is 20 inches long, 12 inches deep and 10 inches high. Find the volume.

30 30 B. Converting Units with Volume Sometimes we need to convert between units of volume. You could convert each of the dimensions first and then calculate the volume using the formula. Another method is to convert units after. Let s take the gas tank example A gas tank is 20 inches long, 12 inches deep and 10 inches high. Find the volume in cubic feet. Note: this can also be done for area calculations, except it is squared. Example: The length and width of a rectangular carpet is 7 ft and 5 ft, respectively. What is the area in square yards? Worksheet: Volume & Converting Units

31 31 (a) Volume and Converting Units Worksheet 1) Determine the volume of each rectangular prism, and then convert the units to the indicated units. in 3 (b) (c) yd 3 (d) ft 3 mi 3

32 32 2) Convert the following measurements of volume or area (be careful whether it is cubed or squared). a) 45 in 3 = ft 3 b) 210 yd 3 = ft 3 c) ft 2 = mi 2 d) 2 m 3 = cm 3 e) 650 in 2 = yd 2 f) 3.25 cm 3 = mm 3 3) A garden bed is 4 by 3 and a 6 layer of soil will be spread over the garden. A bag of soil contains 2 ft 3 of soil. How many bags are needed to cover the garden? 4) A cylindrical shipping tube is 48 inches tall and 6 inches in diameter. What is the surface area in square feet? Answers: 1) (a) 6.1 in 3 (b) yd 3 (c) 0.18 ft 3 (d) mi 3 2) a) ft 3 b)5670 ft 3 c) mi 2 d) cm 3 e) 0.5 yd 2 f) 3250 mm 3 3) 3 bags 4) 6.68 ft 2

33 33 C. Capacity The of a container is the amount it can hold. is the volume of a container. is often used with. In the SI, the basic unit of capacity is the. A is millilitres. Capacity Unit Conversions In imperial units, capacity is measured in gallons. The gallon has two different sizes: The British (UK) gallon: The American (US) gallon:.. The two gallons can be related:.. In measuring liquids for recipes, the US system is often used:

34 34 Examples: Covert the following: (a) 200 fl oz (US) = ml (b) 2500 ml = L (c) 500 ml = cups (d) 12.5 ml = teaspoons (e) 20 gallons (UK) = gallons (US) (f) 5 gallons (US) = ml Worksheet: Converting Capacity

35 35 Converting Capacity Worksheet 1) Convert the following measurements: a) 16 US gallons = litres b) 6 quarts = fl oz c) 55 L = gallons (US) d) 20 fl oz (US) = ml e) 565 ml = fl oz (British) f) 12 gallons (UK) = gallons (US) g) 1500 L = gallons (US) h) 600 fl oz (UK) = ml

36 36 2) Serena is travelling through the US and her car s gas tank has a capacity of 55 litres. a) How much is this in American gallons? b) If gas costs $2.99/gal in Bellingham, WA, how much will it cost to fill her car (assuming it is completely empty)? c) If Serena had the same car in London, England, where gas costs $9.86/gal (converted from pounds), how much will it cost to fill her tank? Calculate the amount of British gallons first. 3) Paula is opening a French bakery and wants to make authentic French recipes. All the recipes are given in metric units, but she has imperial measuring devices. The crème brulée recipe requires 500 ml of cream and 1.25 ml of vanilla. a) How much cream will she need, in cups? b) How much vanilla will she need, in teaspoons? c) How much cream will she need, in fluid ounces? Answers: 1) a) 61.5 L b) 192 fl oz c) 14.3 gal d) ml e) 19.9 f) 14.4 gal g) 390 gal h) ml 2) a) 14.3 gal b) $42.76 c) $ ) a) 2 cups b) 0.25 tsp c) 16.9 fl oz 3.4 Quiz