1

Concepts of Measurement

CHAPTER 11

Slide 2

Chapter

11.1 Linear Measure

11.2 Areas of Polygons and Circles

11.3 The Pythagorean Theorem and the Distance Formula

11.4 Surface Areas

11.5 Volume, Mass, and Temperature

11

Concepts of Measurement

Slide 3

NCTM Standard: Measurement

All students will recognize that objects have attributes that are

measurable explore length, weight, time, area, and

volume (grades 3-5) learn about area, perimeter,

volume, temperature, and angle measure learn both customary and metric systems know rough equivalences between the metric

and customary systems.

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11-1 Linear Measurement The English System

Dimensional Analysis (Unit Analysis)

The Metric System

Distance Properties

Distance Around a Plane Figure

Circumference of a Circle

Arc Length

Slide 5

The English System

UnitEquivalent in Other

Units

yard (yd) 3 feet

foot (ft) 12 inches

mile (mi) 1760 yd or 5280 ft

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Dimensional Analysis (Unit Analysis)

1 5280 and

3 1

yd ft

ft mi

Dimensional Analysis: a process to convert from one unit of measurement to another.

Works with unit ratios (ratios equivalent to 1)

Slide 7

Example:

Which is faster, 50 miles per hour or 50 feet per second?

Answer:

50 mph = 73.3 fps73.3 fps > 50 fpsTherefore, 50 mph is faster than 50 fps

sec

ft3.73

sec 60

min 1

min 60

hr 1

mi 1

ft 5280

hr

mi50

hr

mi50

Slide 8

Example:

Convert:

219 ft = __________ yd

Answer:

yd 73ft 3

yd 1ft 219ft 219

73

Slide 9

mi 79.4ft 5280

mi 1

yd 1

ft 3yd 8432yd 8432

Example:

Convert:

8432 yd = __________ mi

Answer:

4.79

Slide 10

ft 1056mi 1

ft 5280mi 2.0mi 2.0

Example:

Convert:

0.2 mi = __________ ft

Answer:

1056

Slide 11

yd 78.1ft 3

yd 1

in 12

ft 1in 64in 64

Example:

Convert:

64 in = __________ yd

Answer:

1.78

Slide 12

Prefix Symbol Factor

kilo h 1000

hecto h 100

deka da 10

deci d 0.1

centi c 0.01

milli m 0.001

The Metric System

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Unit SymbolRelationship to

Base Unit

kilometer km 1000 m

hectometer hm 100 m

dekameter dam 10 m

meter m base unit

decimeter dm 0.1 m

centimeter cm 0.01 m

millimeter mm 0.001 m

Different units of length in the metric system are obtained by multiplying a power of 10 times the base unit.

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Benchmarks for Metric Units can be used to estimate a meter, decimeter, centimeter, and a millimeter.

Kilometer is commonly used for measuring longer distances: 1 km = 1000 m or nine football fields, including end zones, laid end to end.

Slide 15

Converting Metric Units: are accomplished by multiplying or dividing by power of 10. We simply move the decimal point to the left or right depending on the units.

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Now Try This 11-2 Page 740 If our money system used metric prefixes and the

base unit was a dollar, give metric names to each of the following:

a) dime b) penny c) $10 billd) $100 bill e) $1000 bill

a) dime - decidollarb) penny - centidollarc) $10 bill - dekadollard) $100 bill - hectodollare) $1000 bill - kilodollar

Answer:

Slide 17

Example

Convert:

278 km = _________ m278,000

m 000,278km 1

m 1000km 278km 278

or move the decimal place 3 places to the right.

Answer:

Slide 18

Example

Convert:

278 m = _________ cm2.78

cm 78.2cm 100

m 1m 278m 278

or move the decimal place 2 places to the left.

Answer:

Slide 19

Example

Convert each of the following:

278 mm = ________m0.278

m 278.0mm 1000

m 1mm 278mm 278

Answer:

or move the decimal place to the left 3 places.

Slide 20

Distance Properties

1. The distance between any two points A and B is greater than or equal to 0, written (AB 0).

2. The distance between any two point A and B is the same as the distance between B and A, written (AB = BA).

3. For any three points, A, B, and C, the distance between A and B plus the distance between B and C is greater than or equal to the distance between A and C, written (AB + BC AC).

Slide 21

Triangle Inequality

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AB + BC > AC

Can a triangle be made with sides that are 15 cm, 18 cm, and 37 cm?

No: 15 + 18 = 33

33 is less than 37

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Now Try This 11.3 Page 742 If two sides of a triangle are 31 cm and 85 cm long and the

measure of the third side must be a whole number of centimeters,

a) What is the longest the third side can be?Answer:31 + 85 = 116 – 1 = 115 cm because (116 – 31) = 85

b) What is the shortest the third side can be?Answer:85 – 31 = 54 + 1 = 55 cm because (55 + 31 = 86)

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Distance Around a Plane Figure

Perimeter – the length of a simple closed curve, or the sum of the lengths of the sides of a polygon.

Perimeter has linear measure.

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Example:How many feet of molding are needed to go around the entire room?

Answer:

10 + 12 + 18 + 7 = 47 feet(10 + 7) + (18 – 12) = 17 + 6 = 11 feet47 + 11 = 58 feet

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Circumference of a CircleCircle – the set of all points in a plane that are the same

distance from a given point, the center.

Circumference – the perimeter of a circle.

22 1 3 3.14

7 72C d r

p

p p

» Þ Þ

Þ Þ

Pi – (π) the ratio between the circumference of a circle and the length of its diameter.

d = diameter

r = radius

p = π = pi

C = circumference

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ExampleFind:a. The circumference of a circle with radius 10 m.

Answer:

b. The radius of a circle with circumference 18π ft.Answer:

C = πd or C = 2 π rC = (2)(π)(10)C = 20 π mC ≈ 62.8 m

The length of the diameter (d) is twice the radius (r)

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Arc Length

180

rp q=

r = radius

p = π = pi

q = 0°

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ExampleFind:a. The length of a 36° arc of a circle with diameter 5 inches.

Answer:

b. The radius of an arc whose central angle is 54 ° and whose arc length is 15 cm.Answer:

( )(2.5)(36)1.57

180 180

rql

cm 1692.1556.169

2700

)54)((

)180)(15(

180

q

lr

Radius is ½ the diameter

Slide 29

HOMEWORK 11-1

Pages 746 – 748

# 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27