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Part 3 Module 6 Units of Measure in Geometry Part 3 Module 6 Units of Measure in Geometry Next Tuesday all four parts of your finals week homework project will be posted. We will discuss the details at that time. The first of the four parts will cover geometry (Part 3 Modules 7 through 9), the topic that we will cover over the last two weeks of class. As a preliminary to discussing geometry, we spend a moment in Part 3 Module 6, reviewing some basic facts about units of measure. Part 3 Module 6 will be included on Test 3. Linear Measure Linear measure is the measure of distance. For instance, lengths, heights, and widths of geometric figures are distances, as are the radius, diameter and circumference of a circle. The perimeter of a figure is another example of distance. Distance is measured in linear units, such as inches, feet, yards, miles, meters, centimeters, millimeters and kilometers. Converting Linear Units In the United States, any educated person should be aware of the following relationships among basic units of linear measure in the English system (inches, feet, yards, miles), and among the basic units of linear measure in the metric system (meters, centimeters, millimeters, kilometers). 1 foot = 12 inches 1 yard = 3 feet 1 yard = 36 inches 1 mile = 5280 feet 1 meter = 1000 millimeters 1 meter = 100 centimeters 1 kilometer =1000 meters Multiply or divide Referring to the conversion factors on the previous page, we convert from larger units to smaller units by multiplying (for instance, to convert from yards to inches we multiply by 36); we convert from smaller units to larger units by dividing (for instance, to convert from centimeters to meters we divide by 100). Examples 1. How many meters are in 3 kilometers? 2. How many miles are in 20,000 feet? Answers 1. To convert 3 kilometers to meters we multiply by 1000. 3 x 1000 = 3000 m in 3 km 2. To convert 20000 feet to miles we divide by 5280. 20000/5280 = 3.88 miles in 20000 ft. Square Measure Square units (such as square inches or square centimeters) are used to describe the area of a two-dimensional figure. Area is the amount of 2-dimensional space covered by a flat object. To understand the difference between linear measure and square measure, and to correctly convert between square units, you must realize that one square unit is the area of a square that is 1 unit long and 1 unit wide. Exercise #1 (From The BIG UNIT-izer www.math.fsu.edu/~wooland/unitizer.html ) How many square inches are in 9 square feet? A. 0.75 B. 108 C. 0.063 D. 1296 Solution #1 How many square inches are in 9 square feet? A. 0.75 B. 108 C. 0.063 D. 1296 The answer is not B, because the question is not asking for the number of inches in 9 feet. Square inches and square feet are different from inches and feet. To answer correctly, we must first find the correct conversion factor when converting between square feet and square inches. Solution #, page 2 We must find the correct conversion factor when converting between square feet and square inches. This will depend upon the fact that one foot equals 12 inches, and also upon the meaning on “square.” “One square foot” means literally one foot times one foot. 1 sq. ft = 1 ft. x 1 ft. = 12 in. x 12 in = 144 sq. in. We have figured out that to convert between square feet and square inches, the correct factor is 144, not 12. So, to convert 9 sq. feet to sq. yards, we multiply by 144. 9 x 144 = 1296. The correct choice is D. Illustration The figure below illustrates why 1 square foot = 144 square inches. One foot equals 12 inches 1 foot 1 square foot 1 square inch One square foot equals 144 1 foot square inches 1 foot Cubic Units Cubic units (such as cubic meters or cubic yards) are used to describe the volume of a three-dimensional figure. Volume is the amount of 3-dimensional space occupied by a solid object, for instance, or the amount of fluid that can be contained in a hollow vessel. Exercise #2 How many cubic yards are in 150 cubic feet? A. 16.667 B. 450 C. 50 D. 5.556 Solution #2 How many cubic yards are in 150 cubic feet? A. 16.667 B. 450 C. 50 D. 5.556 The answer is not C, because we are not converting feet to yards. We are converting cubic feet to cubic yards. We must find the correct factor for converting between cubic yards and cubic feet. This will depend upon the fact that one yard equals three feet, and the meaning of the word “cubic.” Solution #2, page 2 “One cubic yard” means literally one yard times one yard times one yard. 1 cubic yard = 1 yard x 1 yard x 1 yard = 3 feet x 3 feet x 3 feet = 27 cubic feet We have figured out that the correct conversion factor is 1 cubic yard = 27 cubic feet. So, to convert 150 cubic feet to cubic yards, we divide by 27. 150/27 = 5.556 The correct choice is D. Illustration The figure below illustrates why 1 cubic yard equals 27 cubic feet.