# 2. Length, Area, and Volume

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1 Name Date Class TEACHING RESOURCES BASIC SKILLS 2. In 1960, the scientific community decided to adopt a common system of measurement so communication among scientists would be easier. The system they agreed to use is called the International System of Units (SI). Scientists all over the world use SI. When you are in the laboratory, all of your measurements will be taken using SI. On this and the following pages you will learn about SI. When you have finished you will be able to 1. identify the SI base unit of length, 2. identify the most commonly used prefixes, and 3. calculate length, area, and volume using the SI base unit for length and units derived from that base unit. Length SI is a decimal system. Our money system is a decimal system: 10 2, or 100, cents to the dollar. This means that all relationships between units of measurement are based on powers of 10. There are several prefixes that can be combined with an SI unit. These prefixes change the unit to mean some other power of 10. For example, the SI base unit of length is the meter. A meter is equal to 39.4 inches in the English system which we use in the United States. When the prefix kilo is added to the base unit meter, the word kilometer is formed (a derived unit meaning 1000 meters). When the prefix centi is combined with the base unit meter the word centimeter is formed (a derived unit meaning 1/100 of a meter). In Figure 1, you will see two rulers that compare the metric scale of measurement with the English scale. Figure 1a illustrates that exactly 1 inch is equal to 2.54 centimeters. Figure 1b illustrates that exactly 1 centimeter is equal to inches. Figure cm 1 cm 0 1 cm cm in in. HOLT BIOSOURCES / Teaching Resources: Basic Skills 1

2 Table 1 lists some commonly used prefixes and their meanings. The prefixes modify the value of the base to describe very small lengths or very long distances, like the diameter of an animal cell or distances between cities. Cells are measured in micrometers, which is abbreviated µm. Distances between cities are measured in kilometers, which is abbreviated km. Find these prefixes in Table 1, and notice their relationship to the base unit. Table 1 Metric Prefixes Prefix Symbol Exponential Rational Meaning tera T ,000,000,000,000 trillion giga G ,000,000,000 billion mega M ,000,000 million kilo k ,000 thousand hecto h hundred deka da ten deci d /10 one tenth centi c /100 one hundredth milli m /1,000 one thousandth micro µ /1,000,000 one millionth nano n /1,000,000,000 one billionth pico p /1,000,000,000,000 one trillionth Because SI is based on powers of 10, we can easily convert from one derived unit to another. For example, there are 100 centimeters in a meter and there are 10 decimeters in a meter. Therefore, there are 10 centimeters in 1 decimeter. Find the answers to questions 1 through 6 using Table 1. Practice Exercise 1 1. What prefix and unit represents 10 meters? 2. What derived unit represents 1/100 of a meter? 3. There are 1,000 millimeters in 1 meter (1 millimeter = 1/1,000 of a meter). Express 500 millimeters in meters. 4. The same measurement is often represented using different prefixes, depending on the purpose of the measurement. For example, the distance between Earth and the sun at its nearest position is usually written as 145 million kilometers. But the same measurement can be represented in meters to 2 HOLT BIOSOURCES / Teaching Resources: Basic Skills

3 Name Date Class emphasize the great distance. The sun is 145 billion meters from Earth at its nearest position in its annual orbit! The same distance can even be stated in terms of gigameters. How many gigameters is 145 million kilometers? 5. The length of line segment A can be represented in millimeters, centimeters, and meters. All of the measures are equivalent, and all represent the length of line segment A. Estimate the length of segment A, and then measure it. Express the measurement in each of the following units. A a. millimeters b. centimeters c. meters 6. Animal and plant cells are studied under a microscope. Some cells are so small that their parts are measured in micrometers. What part of a meter is a micrometer? 7. A millimeter is the smallest distance pictured on the metric ruler in Figure 2. Use the ruler to answer each of the following. a. A points to mm or cm b. B points to mm or cm c. C points to mm or cm d. D points to cm or mm e. E points to cm or mm Figure 2 Area 0 1 cm 2 A B C D E Area represents a measurement of two dimensions, length and width. The area of a region is the number of square units required to cover the region, without overlapping the squares, or allowing gaps. Figure 3a is a square measure 1 centimeter on a side that has an area of 1 square centimeter, denoted by 1 cm 2. Figure 3b is a square measure 2 centimeters on a side that has an area of 4 square centimeters. HOLT BIOSOURCES / Teaching Resources: Basic Skills 3

4 Figure 3 2 cm 1 cm 1 cm 1 cm 2 cm cm 1 cm cm 1 cm 2 4 cm 2 One way to measure area is to count the units contained in a given region. For example, suppose the square in Figure 4a is 1 square unit. The area of Figure 4b can be found by counting each of the same square units contained in Figure 4b. There are 30 non-overlapping squares and no gaps, so the area of Figure 4b is 30 square units. To find the area of very large regions it may not be possible to count each unit. We can find the area of a region by counting the number of units on two adjacent sides. Figure 4b contains 5 rows of 6 squares each, or 5 6 = 30 squares. The area of any rectangle can be found by multiplying the length of any two adjacent sides (length and width). Area = length width A = lw If the length and width are given in centimeters, then the area is calculated in square centimeters (cm 2 ). If the length and width are given in kilometers, then the area is calculated in square kilometers (km 2 ). Both length and width must be stated in the same unit to calculate area. Figure 4 1 unit width 1 unit length 5 units width 6 units length 1 square unit 30 square units 4 HOLT BIOSOURCES / Teaching Resources: Basic Skills

5 Practice Exercise 2 Name Date Class 1. A book measures 11 cm by 18 cm. How much area does the book occupy on the top of a desk? Draw a picture of the book. Label the length of each side and show your work. 2. Maria must replace a worn section of carpet. The damaged area measures 3 m long and 2 m wide. How much carpet does Maria need to buy? 3. A farmer wants to plant half of a field in corn and the other half in cotton. Each half of the field measures 50 m by 100 m. What is the total area the farmer wants to plant? Remember that every measurement of length may be expressed using more than one prefix. Usually, the simplest expression is best. For example, it is better to state the diameter of a cell as 5 µm rather than m,even though the two measurements are equivalent. The distance between cities is more appropriately stated in km rather than cm. 4. Choose the most appropriate metric unit for measuring each of the following: a. Area of a sheet of notebook paper b. Area of a desktop c. Area of a classroom floor d. Area of an airport runway 5. A rectangular plot of land is to be seeded with grass. The plot to be seeded is 22 m by 28 m. If 1 kg of seed is needed for each 85 m 2 of land, how many kg of grass seed are needed? HOLT BIOSOURCES / Teaching Resources: Basic Skills 5

6 Volume As area is used to describe a two-dimensional region, volume represents a measure of a three-dimensional figure. Volume is how much space an object occupies. Volume can be shown with closely stacked cubes that fill a certain space, and have no gaps between them. Units of volume are based on cubes and are called cubic units. As shown in Figure 5, the volume of a rectangular solid can be measured by determining how many cubes cover the base and by counting how many layers of these cubes are needed to reach the height of the object. Figure 5 height 1 1 length height 5 base layer 1 width 8 width 4 length 5 layers As shown in Figure 5a, a unit cube measures 1 unit long, 1 unit wide, and 1 unit high, and equals 1 cubic unit. In Figure 5b, there are 8 4, or 32, cubes in the base, and there are 5 such layers. Therefore, the volume of the object is 8 4 5, or 160, cubic units. The volume of a rectangular solid can be found by multiplying the measures of length, width, and height. Volume = length width height V= lwh If the length, width, and height are given in centimeters, then the volume is calculated in cubic centimeters (cm 3 ). If the length, width and height are given in meters, then the volume is calculated in cubic meters (m 3 ). The length, width, and height must all be stated in the same unit before volume can be calculated. For example, to calculate the volume of a cube that is 2 dm long, 20 cm wide and 0.20 m high, each measurement must be converted to an appropriate common unit. Converting all of the units given, the length is 20 cm, the width is 20 cm and the height is 20 cm. 6 HOLT BIOSOURCES / Teaching Resources: Basic Skills

7 Practice Exercise 3 Name Date Class 1. Calculate the volume of a cube that is 20 cm on a side. 2. What is the volume of a box 24 cm long, 12 cm wide, and 7 cm high? 3. Calculate the volume of a box which is 20 cm wide, 15 cm high and 78 cm long. In the metric system, cubic units may be used for measuring the volume of space that a solid or liquid object occupies. Units such as liters and milliliters are usually used for capacity (the amount a container will hold) or liquid measures. By definition, 1 liter, symbolized by L, equals, or is the capacity of, 1 dm 3. The liter is not an SI unit but is derived from other units. Because 1 L equals 1 dm 3 and 1 dm 3 equals 1000 cm 3, then 1 cm 3 equals 1 ml (1 cm 3 = 1 ml). Figure 6 = = 10 cm 3 1 cm 3 1 ml 10 ml 4. A waterbed is 180 cm wide, 210 cm long, and 20 cm deep. How many liters of water can it hold? 5. To understand the advantages of the metric system, try finding the volume of a waterbed in gallons using the following measurements: 6 ft long, 4.5 ft wide, and 9 in. deep. A byte is not an SI unit, but you will see metric prefixes in many familiar phrases.the words formed may or may not be used to express a precise measurement, but the meaning of the prefix remains the same. See if you can answer the following questions. 6. How many bytes are in a gigabyte? 7.How many bucks are in a megabuck? HOLT BIOSOURCES / Teaching Resources: Basic Skills 7

8 Figure 7 Activity Irregular stone 300 ml water? ml water and stone What is the Volume of the Stone? Because we know that 1 ml is equal to 1 cm 3 in volume, we are able to find the volume of solid objects that are irregular in shape and cannot be measured accurately. One way to do this is to submerge the object in a liquid such as water and to measure the volume of water displaced by the object. Finding the Volume of an Irregular Solid Supplies: One 500 ml graduated cylinder 1 stone about 3 cm in diameter 300 ml of water Pour the 300 ml of water into the 500 ml graduated cylinder, and record 300 ml as your beginning volume of water. Place the stone into the graduated cylinder of water, and record the new level of water in the graduated cylinder. Starting volume Ending volume Difference in volume 1. What is the difference between the first volume of water and the level of water in the beaker after the stone was added, in milliliters? 2. What is the difference in cubic centimeters? 3. What does your finding represent? 4. Explain how you know. 8 HOLT BIOSOURCES / Teaching Resources: Basic Skills

9 Name Date Class Review 1. Why did the world s scientific community decide to adopt a common system of measurement? 2. Explain what is meant by decimal system. 3. Name two other measurements that can be taken using the SI base unit for length and its derived units. 4. Name three units derived from the SI base unit for length, and explain their relationship to the base unit. 5. Convert each of the following. a. 2.4 kilometers = meters b. 436 millimeters = meters 6. Draw line segments that you estimate to be of the following lengths. Then check your estimates with a metric ruler. a. 10 cm c. 1.5 dm b. 29 mm d. 3 mm 7. Find the area of a rectangle 6 cm wide and 0.3 m long. 8. A rectangular piece of land is 1,250 m by 1,140 m. What is the area in square kilometers? 9. An Olympic pool is 50 m long and 25 m wide. If it is 2 m deep throughout, how many liters of water does it hold? 10. An unusual animal cell is being studied under a microscope. It is cubic in shape and is 6.5 µm on a side. What is the volume of the cell? HOLT BIOSOURCES / Teaching Resources: Basic Skills 9

10 Answer Keys HOLT TEACHING RESOURCES ANSWER KEYS 2 Practice Basic Skills Worksheets Exercise 1 1. dekameter (dam) 2. centimeter (cm) m (1/2 meter) Gm 5. a. 87 mm b. 8.7 cm c m 6. 1 millionth, 10 6, 1/1,000, a. 20 mm or 2 cm b. 35 mm or 3.5 cm c. 62 mm or 6.2 cm d. 6.8 cm or 68 mm e. 8.4 or 84mm Exercise 2 1. A = lw; 18 cm 11 cm = 198 cm m 3 m = 6 m m 50 m = 5,000 m 2 2 = 10,000 m 2 4. Accept reasonable answers. Suggested are a. cm; b. m; c. m; d. km m 28 m = 616 m 2 ; 616 m 2 85 m 2 = 7.25 kg of seed Exercise cm 20 cm 20 cm = 8,000 cm cm 12 cm 7 cm = 2,016 cm cm 15 cm 78 cm = 23,400 cm cm 210 cm 20 cm = 756,000 cm 3 = 756,000 ml = 756 L 5. Change inches to feet. (9 in. = 0.75 ft) Find the volume in cubic feet. (6 ft 4.5 ft 0.75 ft = ft 3 ) Find the number of gallons of water in a cubic foot. (7.477 gal) Find number of gallons in ft 3. ( = gal) 6. 1 billion; million; 10 6 HOLT BIOSOURCES / Teaching Resources: Answer Keys 1

11 ANSWER KEYS continued What is the Volume of the Stone? 1. Answers will vary. 2. Answers will vary. 3. The volume of the stone. 4. The stone displaces its own volume in water. The water will rise in the measuring instrument an equivalent amount to the volume of the stone. The level of the water rises in a regular manner that is easy to measure, unlike the irregularly shaped stone. Review 1. To make communication among scientists easier. 2. A decimal system is one based on ten and powers of ten. 3. area and volume 4. kilometer = 1,000 meters; centimeter = 1/100 of a meter; millimeter = 1/1,000 of a meter 5. a. 2,400 meters b meters 6. Answers will vary. 7. Because 0.3 m = 30 cm, 6 cm 30 cm = 180 cm ,250 m = 1.25 km; 1,140 m = 1.14 km; 1.25 km 1.14 km = km m = 5,000 cm; 25 m = 2,500 cm; 2 m = 200 cm 5,000 cm 2,500 cm 200 cm = 2,500,000,000 cm 3 2,500,000,000 cm 3 1,000 = 2,500,000 cm 3 1,000 cm 3 = 1 L 2,500,000 cm 3 = 2,500,000 L m 6.5 m 6.5 m = m 3 2 HOLT BIOSOURCES / Teaching Resources: Answer Keys

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