Measurement: Length, Area, and Volume 10.1 The Measurement Process 10.2Area and Perimeter 10.3The Pythagorean Theorem 10.4Volume 10.5Surface Area Copyright

Measurement: Length, Area, and Volume

10.1 The Measurement Process10.2 Area and Perimeter10.3 The Pythagorean Theorem10.4 Volume10.5 Surface Area

Copyright © 2012, 2009, and 2006, Pearson Education, Inc.

10.1

The Measurement Process

Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 10-2

THE MEASUREMENT PROCESS

1. Choose the property (length, area, volume, etc.) to be measured.

2. Select a unit of measurement.

3. Compare the size of the object with the size of the unit.

4. Express the measurement as the number of units used.


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.


EARLY MEASUREMENT

Units of measurement originally were defined for convenience rather than accuracy.


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.


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


CUSTOMARY UNITS OF AREA

21 ft

2

2 2

1 ft

12 in 144 in


CUSTOMARY UNITS OF AREA

21 yd

2

2 2

1 yd

3 ft 9 ft


CUSTOMARY UNITS OF VOLUME

31 ft

3

3 3

1 ft

12 in 1728 in


CUSTOMARY UNITS OF VOLUME

31 yd

3

3 3

1 yd

3 ft 27 ft


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


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.


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)


THE SI DECIMAL PREFIXES

PREFIX 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 0.000001= 10-6 (mu)


Example 10.3: Changing Units in the Metric System

Convert each of these measurements to the unit shown:

a. 1495 mm = ________ m

b. 29.5 cm = _________ mm

c. 38.741 m = ________ km


1.495

294

38.741

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. Nickelb. Compact automobilec. Two-liter bottle of sodad. Recommended daily allowance of vitamin B-6e. Size D batteryf. Large watermelon

List: 2 mg, 2 kg, 100g, 120kg, 9 kg, 5g


5 g1200 kg

2 kg2 mg

100 g9 kg

TEMPERATURE

FAHRENHEIT SCALE

CELSIUS SCALE

freezing pt of water 32 0boiling pt of water 212 100

10032

180C F 180

32100

F C


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?


mi60

hr

mi 5280 ft 1 hr 1 min= 60

hr 1 mi 60 min 60 sec

60 5280 ft=

60 60 sec

ft= 88

sec

10.2

Area and Perimeter


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.


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.

AREA OF A PARALLELOGRAM

A parallelogram of base b and altitude h has area A given by A = bh.


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..

Example 10.9: Using the Parallelogram Area Formula

Find the area of the parallelogram.


The area is A = (30 cm)(12 cm) = 36 cm2..

AREA OF A TRIANGLE

A triangle of base b and altitude h has area A given by A = ½ bh.


Example 10.10: Using the Triangle Area Formula

Find the area of the triangle.


Using the formula 2110 cm 7 cm 35 cm .

2A

AREA OF A TRAPEZOID

A trapezoid with bases of length a and b and altitude h has area A = ½(a + b)h.


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.


DEFINITION:

The ratio of the circumference C to the diameter d of a circle is .

Therefore,

and .C

C dd


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. 3.14 b. 3.1416


a. (3.14)(7926 miles) = 24,887.64 miles

b. (3.1416)(7926 miles) = 24,900.322 miles

The two different approximations for pi account for the distance of about 12.7 miles in the answers.

DEFINITION:AREA OF A CIRCLE

The area A enclosed by a circle of radius r is 2.A r


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.

p

Example 10.15: continued


Larger pizza =

Smaller pizza =

2 27 in 49 in

2 25 in 25 in

The ratio of areas is 2 249 in / 25 in 1.96.

The 14-inch pizza has about twice the ingredients of the 10-inch pizza.

10.3

The Pythagorean Theorem


THE PYTHAGOREAN THEOREM

If a right triangle has legs of length a and b and its hypotenuse has length c, then

2 2 2.a b c


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.

2 2 2.a b c


THE CONVERSE OFTHE 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.

2 2 2a b c


Example 10.16: Using the Pythagorean Theorem

Find the lengths x and y in the figure.

2 2 213 37

169 1369

1538

x


1538 39.2.x Therefore,

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. 10, 5, 5 3


a. 82 + 152 = 64 + 225 = 289 = 172

The lengths are the sides of a right triangle.

b. The lengths are the sides of a right triangle.

22 25 + 5 3 25 25 3 25 75 100 10

10.4

Volume


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.


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.


Find the volume of the juice can.


The height is h = 9.5 cm, so the volume V = Bh =

2 22.75 cm 7.5625 cm .

3 371.84375 cm , or about 226 cm .

The area of the circular base of the juice can is

VOLUME OF A GENERAL PRISM OR CYLINDER

A prism or cylinder of height h and base area of B has volume

.V Bh


VOLUME OF A PYRAMID

V Bh

A cube can be dissected into three congruent pyramids. 1

3V Bh

Volume of a cube.


VOLUME OF A PYRAMID OR CONE

The volume of a pyramid or cone of height h and base area of B is given by

1.

3V Bh


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

2 3153.361 m 147 m 2,614.689 m .

3

VOLUME OF A SPHERE

The volume of a sphere of radius r is given by

34.

3V r


10.5

Surface Area


DISSECTING A RIGHT PRISM

Two congruent bases


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?




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 cm2 + 1200 cm2 = 2800 cm2

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

2 .SA r rs


SURFACE AREA OF A SPHERE

The surface area of a sphere of radius r is given by

24 .S r


Documents

Measurement: Length, Area, and Volume 10.1 The Measurement Process 10.2Area and Perimeter 10.3The Pythagorean Theorem 10.4Volume 10.5Surface Area Copyright