CHAPTER 6: MEASUREMENTS

College Prep Essential Math Chapter 6: Measurements

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CHAPTER 6: MEASUREMENTS

Chapter Objectives

By the end of this chapter, students should be able to:

Identify terms used in measurement in both metric and U.S. customary systems

Distinguish between 1, 2, and 3 dimensional measures. Convert U.S. customary unit measures. Convert between U.S. and metric systems. Solve applications of unit measurements.

Contents CHAPTER 6: MEASUREMENTS .................................................................................... 1

SECTION 6.1 MEASUREMENT .................................................................................. 2

A. U.S. CUSTOMARY SYSTEM ......................................................................... 2

I. Weight ............................................................................................................. 2

II. Length.......................................................................................................... 3

III. Area ............................................................................................................. 5

IV. Volume ........................................................................................................ 6

B. METRIC SYSTEM .......................................................................................... 9

I. Weight ............................................................................................................. 9

II. Length.......................................................................................................... 9

C. REVIEWING U.S. CUSTOMARY SYSTEM AND METRIC SYSTEM ........... 12

EXERCISES........................................................................................................... 13

SECTION 6.2 CONVERSIONS ................................................................................. 14

A. CONVERT U.S. CUSTOMARY UNITS ......................................................... 14

B. CONVERT METRIC UNITS .......................................................................... 17

C. CONVERT BETWEEN U.S. AND METRIC SYSTEMS ................................ 21

EXERCISES........................................................................................................... 24

CHAPTER REVIEW .................................................... Error! Bookmark not defined.


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SECTION 6.1 MEASUREMENT

We can measure different attributes of an object. Think about a desk. You can measure

how high it is off the ground, how wide the seat is, the area of the desktop, and the space

it takes up in the room. You can measure many things like length, area, volume, and

weight.

In the United States, two main systems of measurement are used: the metric system

and the U.S. customary measurement system. The U.S. Customary System is derived

from the British system of measure and will be familiar to you. The Metric system is more

commonly used around the world.

A. U.S. CUSTOMARY SYSTEM

I. Weight

When you mention how heavy or light an object is, you are referring to its weight. The

tables below list common units of measurement for weight for the U.S. Customary

System.

U.S. Customary Measurement

Ounces (𝑜𝑧)

Pounds (𝑙𝑏)

Tons (𝑡)

The table below describes a unit of weight and provides an example to illustrate the size

of the unit of measurement.

Ton Some species of whales can reach weights of up to 200 tons.

Pound Meat is a product that is usually sold by the pound. One pound of beef makes about four hamburger patties.

Ounces Ounces are used to measure lighter objects. A stack of 11 pennies is equal to about one ounce.


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Media Lesson U.S. Customary Units: Weight (Duration 3:39)

View the video lesson, take notes below.

Unit Items that could be measured in this unit.

Pounds

Ounces

Tons

It is important to note there are different types of tons. You might have heard about the

small ton and the long ton. The small ton refers to the U.S. Customary ton. The long ton

is the British ton. The U.S. Customary ton is 2,000 pounds while the British ton is 2,240

pounds. The metric ton is called the tonne. It is 1,000 kilograms. In this chapter we will

use the small ton.

YOU TRY:

a) Determine what units would be appropriate to use to measure the following weights.

II. Length

Length can be thought of as the distance between two points.

The table below lists common units of measurement of length for the U.S. Customary

System.


Inches (𝑖𝑛)

Foot (𝑓𝑡)

Yard (𝑦𝑑)

Mile (𝑚)

Item U.S. Customary Unit

The weight of an elephant

The weight of bananas

The weight of a lipstick

https://www.youtube.com/watch?v=Dj1rbIP8PHM


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The table below describes a unit of length and provides an example to illustrate the size

of the unit of measurement.

Inches Some people donate their hair to be made into wigs for cancer patients who have lost hair as a result of treatment. One company requires hair donation to be at least 8 inches long.

Feet The length of your bed, couch, and table can be measured in feet.

Yards Soccer fields vary in their size. An official field can be any length between 100 and 130 yards.

Miles A marathon is 26.2 miles long.

Media Lesson U.S. Customary Units: Distance (Duration 5:10)



Feet

Inches

Yard

Mile

YOU TRY:

b) Determine what units would be appropriate to measure the following lengths.


The distance from home to campus

The height of a water bottle

The length of an ant

https://www.youtube.com/watch?time_continue=112&v=M9bisHkXbKc


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III. Area

Area is the amount of space within the boundaries of a 2-dimensional shape. The table

below lists common units of measurement for area.


square inches (𝒊𝒏𝟐) (a square that has 1-inch long sides)

square feet (𝒇𝒕𝟐) (a square that has 1-foot long sides)

square yard (𝒚𝒅𝟐) (a square that has 1-yard long sides)

The table below describes a unit of area and provides an example to illustrate the size of

the unit of measurement.

Square inch A quarter is about 1 inch wide. Imagine drawing a square around a quarter. That square would be about the size of 1 square inch.

Square feet The front of your school binder has an area close to 1 square foot.

Square yard Poster papers are 2 feet by 3 feet. They are a little less than 1 square yard.

The unit you use to measure area depends on the unit you would use to measure the

length of your object. A tile has sides you would measure in inches, so its area would be

measured in square inches.

YOU TRY:

c) What units would be appropriate to use to measure the following?


The floor of your living room

The area of a sheet of paper

The area of a post-it note

The lot size of your home


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IV. Volume

Volume is the space taken up by a 3-dimensional object. The table below lists common

units of measurement.


Teaspoon

Table spoon

Fluid Ounce (𝑓𝑙 𝑜𝑧)

Cup (𝑐)

Pint (𝑝𝑡)

Quart (𝑞𝑡)

Gallon (𝑔)

Cubic inch (𝑖𝑛3) (a cube that has 1-inch long sides)

The table below describes each unit of volume and provides an example to illustrate the

size of the unit of measurement.

Fluid Ounce The amount of liquid medicine is often measured in fluid ounces.

Cup The volume of a measuring cup is one cup.

Pint Cartons of ice cream are often measured in pints.

Quart You often see quarts of milk sold in supermarkets.

Gallon Gas is sold by gallons.

Cubic Inch This is about the size of a stack of 15 quarters.

Media Lesson U.S. Customary Units: Fluid Volume (Duration 7:20)



Teaspoon

Table spoon

Fluid Ounce

https://www.youtube.com/watch?v=eqKnx5Yk508


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Cups

Pints

Quarts

Gallon

Example: The table below lists appropriate units of measurement for each item.


The amount of water in a bathtub gallons

The amount of coffee in a cup cups and cubic inches

The amount of fluid in a drop of water ounces

YOU TRY:

a) What units would be appropriate to measure the following?


The amount of water in a pool

The amount of water in a bottle

The amount of fluid in an allergy shot


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Capacity 1 gallon = 4 quarts = 16 cups 1 quart = 2 pints = 4 cups 1 pint = 2 cups 1 cup = 8 fluid ounces 3 teaspoon = 1 tablespoon


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B. METRIC SYSTEM

The metric system is more commonly used around the world. You may have heard of

kilograms, centimeters, millimeters, and liters. These are units of the metric system. The

metric system uses three basic units: meters, grams, and liters. The meter is the basic

unit of length. The gram is the basic unit of weight. The liter is the basic unit of volume.

I. Weight

Common units of weight of the Metric System are kilograms, grams, and milligrams.

Media Lesson Metric System: Units of Weight (Duration 4:25)



Gram

Kilogram

Milligram

YOU TRY:

a) Determine what units would be appropriate to use to measure the following

weights.

Item Metric Unit

The weight of an elephant

The weight of bananas

The weight of a lipstick

II. Length

Common units of length of the Metric System are kilometers, meters, centimeters, and

millimeters.

Media Lesson Metric System: Units of Distance(Duration 6:55)



Meter

https://www.youtube.com/watch?time_continue=2&v=TD1zuENbEdk

https://www.youtube.com/watch?time_continue=38&v=I3kQJvR7ZIg


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Kilometer

Millimeter

YOU TRY:

b) Determine what units would be appropriate to use to measure the following

lengths.

Item Metric Unit

The distance from home to campus

The height of a water bottle

The length of an ant


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Area

Like the U.S. Customary System units of area in the metric system are square units.

Common units of area of the Metric System are square meters and square centimeters.

If you would measure the length of an object in centimeters, then it’s area would be

measured in square centimeters.

Example:

YOU TRY:

c) What units would be appropriate to use to measure the following?


The floor of your living room

The area of a sheet of paper

The area of a post-it note

The lot size of your home

Volume

Common units of volume of the metric system are kiloliters, liters, and milliliters. Like the

U.S. Customary System volume can also be measured in cubic units. You could measure

an object in cubic centimeters and cubic meters.

Media Lesson Metric System: Units of Volume (Duration 4:13)



Liter

Milliliter

Kiloliter

Item Metric Unit

Surface of a television square centimeters

Area of a fence square meters

The cover of a book square centimeters

https://www.youtube.com/watch?time_continue=141&v=LhMEqsL_M5o


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YOU TRY:

d) What units would be appropriate to measure the following volumes?

Item Metric Unit

The amount of water in a pool

The amount of water in a bottle

The amount of fluid in an allergy shot

C. REVIEWING U.S. CUSTOMARY SYSTEM AND METRIC SYSTEM

The following table compares units of the Metric System to the U.S. Customary System.

Centimeter A centimeter is a little less than half an inch.

Meter A meter is slightly larger than a yard.

Kilometer A kilometer is about one and half of a mile.

Gram 28 grams is about the same as 1 ounce.

Kilogram A kilogram is a little more than 2 pounds.

Liter A liter is slightly larger than a quart.

4 liters is a little more than 1 gallon.

Media Lesson U.S. Customary and Metric Units |Ratios, Proportions, Unit, and Rates (Duration 5:09)


Sort the following units of measurement into two categories: U.S. Customary Units and

Metric Units. Then indicate whether each unit measures length, weight, or volume.

Liter Gallon Decigram Millimeter Foot Kilogram Centiliter

Centimeter Cup Meter Pound Inch Ounce Yard

https://www.youtube.com/watch?v=y2RHWA7t0WM


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EXERCISES

1) Determine what units would be appropriate to use to measure the following

lengths.

Item U.S. Customary Unit Metric Unit

The length of a football field

The height of a textbook

The length of an eyelash

2) For each of the following items, state which U.S. Customary units and which

metric units would be the most appropriate.

a. The weight of a dog

b. The distance from Phoenix to Lost Angeles

c. The area of foundation of a house

d. The length of a lady bug

3) Determine whether the following describe length, area, or volume.

a. The diagonal of a television

b. The screen of a television

c. The bottom surface of a mug

d. The amount of coffee that a mug will hold

4) Which of the following sets of three units are all metric measurements of length?

a. Inch, foot, yard

b. Kilometer, centimeter, millimeter

c. Kilogram, gram, milligram

d. Kilometer, foot, meter

5) Which of the following sets of three units are all measurements of volume?

a. Inch, foot, yard

b. square inches, square feet, square yards

c. cubic inch, cubic feet, liters

d. gram, pound, ounce


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SECTION 6.2 CONVERSIONS

A. CONVERT U.S. CUSTOMARY UNITS

Sometimes we know a measurement in a particular unit, but we are interested in the value

of the measurement in a different unit. Suppose we know that the length of a table is 7ft,

but we want to know what the value of the measurement is in inches. The process of

converting a measurement from one unit to another is called unit conversion.

The table below shows some equivalencies between units.

U.S. System Units Conversions

Length Volume

1 ft = 12 in 1 yd = 3 ft 1 mi = 5,280 ft

1 c = 8 oz 1 pt = 2 c 1 qt = 2 pt 1 qt = 32 oz 1 gal = 4 qt

Weight Time

1 lb = 16 oz 1 ton = 2,000 lb

1 min = 60 sec 1 hr = 60 min 1 day = 24 hr 1 wk = 7 days 1 yr = 52 wk 1 yr = 12 mo 1 yr = 365 days

To convert between units of measure we will be multiplying by a conversion factor

written in fraction form. Conversion factors come from the equivalencies in the table. We

write our conversion factors with the units we want to convert in the denominator. When

we multiply by this conversion factor the units divide out and leave us with a new unit.

Example: Convert 7ft to inches

Looking at our table we see

1 𝑓𝑡 = 12 𝑖𝑛

We write this equivalency in fraction form. Since we are converting 7 ft, feet will be in

the denominator.

12 𝑖𝑛𝑐ℎ𝑒𝑠

1 𝑓𝑜𝑜𝑡

Once you have a conversion factor you will multiply 7 ft by it. Note, you can write 7ft as

a fraction.

7 𝑓𝑒𝑒𝑡

1×


1 𝑓𝑜𝑜𝑡=


1= 84 𝑖𝑛𝑐ℎ𝑒𝑠


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7 feet is multiplied by the numerator of 12 inches and is then divided by the denominator

of 1 foot. The unit of feet is divided out and we are left with inches.

Media Lesson Single Step Unit Conversion (US Standard – Unit Fractions) (Duration 4:18)

View the video lesson, take notes and complete the problems below.

a. 4 lb = _________ oz

b. 10 yd = _________ ft

c. 2.4 pt = _________ c

Media Lesson Converting Between Different Units of Weight in Standard or American System (Duration 2:32)


Convert the measurement. Round to the nearest tenth if needed.

240 oz = _________ lbs

3 T = _________ lbs = _________ oz

YOU TRY:

Find the conversions below. e) 4 pounds to ounces

f) 10 yards to feet

https://www.youtube.com/watch?v=5xRQQKwecL4&feature=youtu.be

https://www.youtube.com/watch?v=_wxrSBEhLpI

https://www.youtube.com/watch?v=_wxrSBEhLpI


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Sometimes converting from one unit to another may have several steps. We will need to

use several conversion factors.

Example: Convert 66 inches to yards.

We do not have an equivalency between inches and yards. We will first need to convert

inches to feet and then convert our feet to yards.

1 𝑓𝑡 = 12 𝑖𝑛𝑐ℎ𝑒𝑠 1 𝑦𝑑 = 3 𝑓𝑒𝑒𝑡

Use these equivalencies for the conversion factors.

66 𝑖𝑛

1×

1 𝑓𝑡

12 𝑖𝑛×

3 𝑦𝑑

1 𝑓𝑡

Cancel out units.

66 × 1 × 3 𝑦𝑑

1 × 12 × 1

Multiply.

198 𝑦𝑑

12

Simplify.

16.5 𝑦𝑎𝑟𝑑𝑠

66 inches is 16.5 yards.

Media Lesson Multi-Step Conversions (US Standard – Unit Fractions) (Duration 4:48)


Some conversions require more than one step. See how the single-step conversion

process is expanded in each of the following examples.

a) How many minutes are in a week?

b) Bryan needs 10 cups of fruit juice to make Sangria. How many quarts of juice

should he buy at the grocery store?

https://www.youtube.com/watch?v=nU58-tchEQE&feature=youtu.be


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YOU TRY:

Find the conversions below.

g) 9 weeks to minutes h) 250,00 miles to yards

B. CONVERT METRIC UNITS

Convert Using Prefixes

In the metric system, units are related by powers of 10. The root names reflect this

relation. For example the basic unit for measuring length is the meter. One kilometer is

1,000 meters. The prefix kilo- means thousand. One centimeter is 1

100 of a meter. The

prefix centi - means one-hundredth. This makes conversions easy in the metric system.

The table below shows the relationship between different units.


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Metric Chart

KILO HECTO DEKA DECI CENTI MILLI

1,000× 𝑏𝑎𝑠𝑒

100 × 𝑏𝑎𝑠𝑒 10 × 𝑏𝑎𝑠𝑒 Base Unit

. 1 × 𝑏𝑎𝑠𝑒 . 01 × 𝑏𝑎𝑠𝑒 . 001× 𝑏𝑎𝑠𝑒

Kilometer (km)

Hectometer (hm)

Dekameter (dam)

Meter (m)

Decimeter (dm)

Centimeter (cm)

Millimeter (mm)

Kiloliter (kl)

Hectoliter (hl)

Dekaliter (dal)

Liter (l) Deciliter (dl)

Centiliter (cl)

Milliliter (ml)

Kilogram (kg)

Hectogram (hg)

Dekagram (dag)

Gram (g) Decigram (dg)

Centigram (cg)

Milligram (mg)

It is better to remember the relationships between the prefixes and the base unit. These

relationships will help us convert to different units just by moving our decimal point.

Metric Chart


1,000× 𝑏𝑎𝑠𝑒

100× 𝑏𝑎𝑠𝑒

10 × 𝑏𝑎𝑠𝑒 Base Unit . 1 × 𝑏𝑎𝑠𝑒 . 01× 𝑏𝑎𝑠𝑒

. 001× 𝑏𝑎𝑠𝑒

Example: Nick ran a 10-kilometer race. How many meters did he run?

Looking at the table we see km is 3 spaces away from our base unit of meters.

We will move the decimal point by 3 decimal spaces in the same direction to convert to

meters.

Nick ran 10,000 meters.


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Media Lesson Examples: Converting Between Metric Units (Duration 4:30)


Metric Chart


1,000 100 10 Base Unit . 1 . 01 . 001

Convert the measurement.

4 m = __________________ cm

2.5 mm = __________________ m

250 ml = __________________ kl

1.25 kg = __________________ g

YOU TRY:

Convert the metric units. i) 4,200 g = __________________ mg j) 320 ml __________________l k) 1.2 kg = __________________ mg

Convert Using Conversion Factors

The table below lists some equivalencies. We can use these equivalencies to convert

between metric units like we did for the U.S. system of measurement.

Metric Conversions

1 centimeter = 10 millimeters

1 meter = 100 centimeters

1 kilometer = 1,000 meters

https://www.youtube.com/watch?v=6GeUba-Jy8s


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Let us revisit the example from before.

Example: Nick ran a 10-kilometer race. How many meters did he run?

Based on the table 1 kilometer is 1,000 meters.

1 𝑘𝑚 = 1,000 𝑚

Use this equivalency for the conversion factor.

10 𝑘𝑚

1×

1,000 𝑚

1 𝑘𝑚

Cancel out units.

10 × 1,000 𝑚

1

Multiply.

10,000 𝑚

1

Simplify.

10,000 𝑚

Nick rand 10,000 meters.

Media Lesson Introduction to Metric Conversions (Duration 8:30)


a. 4,200 g = __________________ mg

b. 45 cm = __________________ m

c. 7,236,137 ml = __________________ kl

https://www.youtube.com/watch?v=6u1KLltlYoE&feature=youtu.be


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YOU TRY:

Use conversion factors to convert the metric units. l) 3,200 g to kilograms

m) Herman bought a rug 2.5 meters in length. How long is that in centimeters?

C. CONVERT BETWEEN U.S. AND METRIC SYSTEMS

Although the U.S. relies heavily on our standard measurement system, we do use some

metric units. Therefore, we need to know how to move back and forth between the

systems. We will use conversion factors to achieve this process.

A table of some common U.S./Metric conversions are below. Note that many of these

conversions are approximations.

Length 1 mi = 1.61 km 1 yd = 0.9 m 1 in = 2.54 cm

Weight 1 kg = 2.2 lb 1 g = 0.04 oz

Volume 1 L = 1.1 qt 1 gal = 3.8 L 1 L = 2.1 pt

0.621 mi = 1 km 1.094 yd = 1 m 0.394 in = 1 cm

0.424 kg = 1 lb 1 oz = 28.3 g

1 cubic yd = 0.76 cubic m 1 cubic in = 16.4 cubic cm

Like before we will convert between units using conversion factors.


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Media Lesson Examples: Convert Lengths Between the Standard and Metric System (Duration 2:36)



16 in ≈ __________________ cm

25 m ≈ __________________ ft

Media Lesson Examples: Convert Weight Between the Standard and Metric System (Duration 3:21)



180 𝑙𝑏𝑠 ≈ _____________𝑘𝑔

12 𝑘𝑔 ≈ _____________𝑜𝑧

https://www.youtube.com/watch?v=3HumkXBKP0k

https://www.youtube.com/watch?v=B8_U2waOvuM


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YOU TRY:

Find the conversions below. Round to the hundredths place. n) 10 kilometers to miles

o) Express 2 liters in terms of quarts.


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EXERCISES

In the following exercises, convert the U.S. Customary Units.

1) A park bench is 6 feet long. Convert the length to inches.

2) A floor tile is 2 feet wide. Convert the width to inches.

3) Carson is 45 inches tall. Convert his height to feet.

4) Jon is 6 feet 4 inches tall. Convert his height to inches.

5) An empty bus weighs 35,000 pounds. Convert the weight to tons.

6) Rocco waited 11

2 hours for his

appointment. Convert the time to seconds.


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7) Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.

8) April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.

In the following exercises, convert the Metric Units. Use the method you prefer.

9) Ghalib ran 5 kilometers. Convert the length to meters.

10) A bottle of wine contained 750 milliliters. Convert this to liters.

11) The width of the wading pool is 2.45 meters. Convert the width to centimeters.

12) Mount Whitney is 3,072 meters tall. Convert the height to kilometers.

13) June’s multivitamin contains 1,500 milligrams of calcium. Convert to grams.

14) A typical ruby-throated hummingbird weighs 3 grams. Convert this to milligrams.

15) One serving of gourmet ice ream has 25 grams of fat. Convert this to milligrams.

16) A bottle of medicine contained 300 milliliters. Convert to liters.


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In the following exercises, convert the U.S. and Metric System. Round answers to the

hundredths place.

17) Bill is 75 inches tall. Convert his height to centimeters.

18) A 5K run is 5 kilometers long. Convert this length to miles.

19) Jackson’s backpack weighs 15 kilograms. Convert the weight to pounds.

20) Each American throws out an average of 1,650 pounds of garbage per year. Convert this weight to kilograms.

21) Ozzie put 14 gallons of gas in his truck. Convert the volume to liters.

22) A planter holds 1.5 liters of soil. Convert this to gallons.


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Online Quiz

Log on to Canvas to take the section quiz

Directions: It is very useful to save your math exercise work and use it as a chapter

test review when you study for your chapter test and final.

1) Write each question on the screen down to for your record

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mistake.

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to study for your chapter test later.

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CHAPTER 6: MEASUREMENTS

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