1. Measurement: Length, Area, and Volume
10.1 The Measurement Process
10.2 Area and Perimeter
10.3 The Pythagorean Theorem
10.4 Volume
10.5 Surface Area
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2. The Measurement Process
10.1
The Measurement Process
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3. THE MEASUREMENT PROCESS
Choose the property (length, area, volume, etc.) to be measured.
Select a unit of measurement.
Compare the size of the object with the size of the unit.
Express the measurement as the number of units used.
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4. THE MEASUREMENT PROCESS
A measurement is most often only an approximation, and decisions must be made to choose appropriate measurement tools and units to provide accuracy and precision.
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5. EARLY MEASUREMENT
Units of measurement originally were defined for convenience rather than accuracy.
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6. THE U.S. CUSTOMARY SYSTEM
The U.S. customary system of measurement is also known as the “English” system.
Learning the customary system requires extensive memorization.
Using the system involves computations with cumbersome numerical factors.
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7. CUSTOMARY UNITS OF LENGTH
12 inches = 1 foot
3 feet = 1 yard
16.5 feet = 1 rod
660 feet = 1 furlong
5280 feet = 1 mile
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8. CUSTOMARY UNITS OF AREA
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9. CUSTOMARY UNITS OF AREA
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10. CUSTOMARY UNITS OF VOLUME
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11. CUSTOMARY UNITS OF VOLUME
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12. CUSTOMARY UNITS OF CAPACITY
3 teaspoons = 1 tablespoon
1 tablespoon = ½ fluid ounce
8 fluid ounces = 1 cup
4 cups = 1 quart
4 quarts = 1 gallon
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13. THE METRIC SYSTEM
The metric system of measurement is also known as the SI system, after its French name, Systeme Internationale.
The principal advantage of the metric system – other than its universality – is the ease of comparison of units. The ratio of one unit to another is always a power of 10.
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14. FUNDAMENTAL UNITS IN THE METRIC SYSTEM
Length meter (m)
Area square meter (m2)
square kilometer (km2)
Volume cubic centimeter (cm3)
cubic meter (m3)
liter (L)
Weight kilogram (kg)
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15. THE SI DECIMAL PREFIXES
FACTOR
SYMBOL
kilo
1000 = 103 k
hecto
100 = 102 h
deka (or deca)
10 = 101 da
(none for basic unit)
1 = 100
(none)
deci
0.1= 10-1 d
centi
0.01 = 10-2 c
milli
0.001= 10-3 m
micro
= 10-6
 (mu)
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16. Example 10.3: Changing Units in the Metric System
Convert each of these measurements to the unit shown:
a mm = ________ m
b cm = _________ mm
c m = ________ km
1.495
294
38.741
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17. Example 10.4: Estimating Weights in the Metric System
Match each term to the approximate weight of the item taken from the list that follows:
a. Nickel
b. Compact automobile
c. Two-liter bottle of soda
d. Recommended daily allowance of vitamin B-6
e. Size D battery
f. Large watermelon
List: 2 mg, 2 kg, 100g, kg, 9 kg, 5g
5 g
1200 kg
2 kg
2 mg
100 g
9 kg
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18. TEMPERATURE freezing pt of water 32 0 boiling pt of water 212 100
FAHRENHEIT
SCALE
CELSIUS SCALE
freezing pt of water
32 0
boiling pt of water
212
100
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19. Example 10.5: Computing Speeds and Capacity with Units
A cheetah can run 60 miles per hour. What is the speed in feet per second?
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20. 10.2 Area and Perimeter Slide 10-20
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21. AREA OF A REGION IN THE PLANE
Let R be a region and assume that a unit of area is chosen. The number of units required to cover a region in the plane without overlap is area of the region R.
Usually, squares are chosen to define a unit of area, but any shape that tiles the plane can serve equally well.
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22. The are of a rectangle is the product of its length and width.
AREA OF A RECTANGLE
A rectangle of length l and width w has area A given by the formula A = lw.
The are of a rectangle is the product of its length and width.
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23. AREA OF A PARALLELOGRAM
A parallelogram of base b and altitude h has area A given by A = bh.
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24. Example 10.9: Using the Parallelogram Area Formula
Find the area of the parallelogram.
The base is 10 cm and the height is 4 cm, so the area is A = (10 cm)(4 cm) = 40 cm2..
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25. Example 10.9: Using the Parallelogram Area Formula
Find the area of the parallelogram.
The area is A = (30 cm)(12 cm) = 36 cm2..
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26. A triangle of base b and altitude h has area A given by A = ½ bh.
AREA OF A TRIANGLE
A triangle of base b and altitude h has area A given by A = ½ bh.
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27. Example 10.10: Using the Triangle Area Formula
Find the area of the triangle.
Using the formula
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28. AREA OF A TRAPEZOID
A trapezoid with bases of length a and b and altitude h has area A = ½(a + b)h.
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29. Perimeter is a length measurement.
PERIMETER OF A REGION
If a region is bounded by a simple closed curve, then the perimeter of the region is the length of the curve. More generally, the perimeter of a region is the length of its boundary.
Perimeter is a length measurement.
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30. The ratio of the circumference C to the diameter d of a circle is .
DEFINITION: 
The ratio of the circumference C to the diameter d of a circle is .
Therefore,
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31. Example 10.14: Calculating the Equatorial Circumference of the Earth
The equatorial diameter of the earth is 7926 miles. Calculate the distance around the earth at the equator, using the following for pi.
a b
a. (3.14)(7926 miles) = 24, miles
b. (3.1416)(7926 miles) = 24, miles
The two different approximations for pi account for the distance of about 12.7 miles in the answers.
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32. DEFINITION: AREA OF A CIRCLE
The area A enclosed by a circle of
radius r is
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33. Example 10.15: Determining the Size of a Pizza
A 14-inch pizza has the same thickness as a 10-inch pizza. How many times more ingredients are there on the larger pizza?
Pizzas are measured by their diameters. So the radii of the two pizzas are 7-inch and 5-inch, respectively. Since the thicknesses are the same, the amount of ingredients used is proportional to the areas of the pizza.
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34. Example 10.15: continued Larger pizza = Smaller pizza =
The ratio of areas is
The 14-inch pizza has about twice the ingredients of the 10-inch pizza.
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35. The Pythagorean Theorem
10.3
The Pythagorean Theorem
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36. THE PYTHAGOREAN THEOREM
If a right triangle has legs of length a and b and its hypotenuse has length c, then
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37. PROVING THE PYTHAGOREAN THEOREM
The sum of the areas of the squares on the leg of a right triangle is equal to the area of the square on the hypotenuse.
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38. THE CONVERSE OF THE PYTHAGOREAN THEOREM
Let a triangle have sides of length a, b, and c.
If , then the triangle is a right triangle and the angle opposite the side of length c is its right angle.
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39. Example 10.16: Using the Pythagorean Theorem
Find the lengths x and y in the figure.
Therefore,
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40. Example 10.18: Checking for Right Triangles
Determine whether the three lengths given are the lengths of the sides of a right triangle.
a. 15, 17, 18 b.
a = = 289 = 172
The lengths are the sides of a right triangle. b.
The lengths are the sides of a right triangle.
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41. 10.4
Volume
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42. SURFACE AREA AND VOLUME
The surface area of a figure in space measures the boundary of the space figure.
The volume measures the amount of space enclosed within the boundary.
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43. Example 10.19: Computing the Volume of a Right Prism and A Right Cylinder
Find the volume of the gift box.
The base area, B, consists of a square of length 20 cm and four 5-cm-by-20-cm rectangles, so B = 800 cm2. The height is h = 10 cm, so the volume is V = Bh = 8000 cm3, which can also be expressed as 8 liters.
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44. Example 10.19: continued Find the volume of the juice can.
The area of the circular base of the juice can is
The height is h = 9.5 cm, so the volume
V = Bh =
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45. VOLUME OF A GENERAL PRISM OR CYLINDER
A prism or cylinder of height h and base area of B has volume
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46. VOLUME OF A PYRAMID Volume of a cube.
A cube can be dissected into three congruent pyramids.
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47. VOLUME OF A PYRAMID OR CONE
The volume of a pyramid or cone of height h and base area of B is given by
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48. Example 10.20: Determining the Volume of an Egyptian Pyramid
The pyramid of Khufu is 147 m high, and its square base is 231 m on each side. What is the volume of the pyramid?
The area of the base is (231 m)2. Therefore, the volume is
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49. The volume of a sphere of radius r is given by
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50. 10.5
Surface Area
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51. DISSECTING A RIGHT PRISM
Two congruent bases
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52. Example 10.22: Finding the Surface Area of a Prism-Shaped Gift Box
The height of the gift box is 10 cm, the longer edges are 20 cm long, and the short edges of the square corner cutouts are each 5 cm long. What is the surface area of the box?
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53. Example 10.22: continued
Each base has the area B = 800 cm2. The lateral area is that of a rectangle 10 cm high and 120 cm long. That is, the lateral surface area is 1200 cm2. Altogether, the surface area of the box is SA= 2 × 800 cm cm2 = 2800 cm2
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54. SURFACE AREA OF A RIGHT CIRCULAR CONE
Let a right circular cone have slant height s and a base of radius r .
Then the surface area SA of the cone is given by
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55. SURFACE AREA OF A SPHERE
The surface area of a sphere of radius r is given by
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