Section 9.2
Measuring Area and
Volume
1.
Measuring Area
Square unit – A unit with 2 sides (usually
labeled length and width), with each of these
sides being one unit in length.
Objectives
Use square units to measure area.
2.
Use dimensional analysis to change
units for area.
3.
Use cubic units to measure volume.
4.
Use English and metric units to
measure capacity.
9/14/2011
Section 9.2
Section 9.2
2
Square Units of Measure:
The English System
Example: Measuring Area
9/14/2011
What is the area of this region?
1 square foot (ft²) =
1 square yard (yd2) =
144 square inches (in.²)
9 square feet
Solution:
We count 12 square units.
Therefore, the area is 12 square units.
What is the area of the first two rows of this
region?
Solution:
We count 8 square units.
Therefore9/14/2011
the areaSection
is 89.2 square units.
3
Square Units of Measure: The English
System
Example: Measuring Area
What is the area of the region in square yards if
each unit is a square foot?
1 acre (a) = 43,560 ft² = 4840 yd²
Square foot
Solution:
1 1/3 square yards
1 square mile (mi²) = 640 acres
9/14/2011
Section 9.2
5
9/14/2011
Section 9.2
6
Example - Using Square Units to
Compute Population Density
Example - Appropriate units
Name something appropriately measured in
the following units:
square inches, square feet, acres, square
miles
Some possible solutions:
in2: sheet of paper, television screen
ft2: house, room
acres: yard, park
mi2: town or city
Population density measures number of people
per area.
Wikipedia reports that Guttenburg, New Jersey
has the highest population density of any U.S.
city (using 2010 census data), with
approximately 57,924 people per square mile.
The Manhattan borough of New York City has a
population of 1,585,873 and an area of 23
square miles. Is Guttenburg or Manhattan
more densely populated?
Section 9.2
9/14/2011
English and Metric Equivalents for
Area
Example solution
We compute the population density by
dividing Manhattan’s population by its
area in square miles.
1 square inch(in²)
≈ 6.5 square centimeters (cm²)
Population density =
1 square foot (ft²)
≈ 0.09 square meter (m²)
population = 1,585,873 people
area
23 square miles
1 square yard (yd²) ≈ 0.8 square meter (m²)
Using a calculator, we obtain a population
density of approximately 68,951 people
per square mile. Manhattan is more
densely populated that Guttenburg.
1 square mile (mi²) ≈ 2.6 square kilometers (km²)
1 acre
≈ 0.4 hectare (ha)
9/14/2011
Example - Using Dimensional
Analysis on Units of Area
Section 9.2
10
Example Continued
A property in Costa Rica is advertised at
$200,000 for 5.9 hectares.
a) Find the area of the property in acres.
b) Find the price per acre
Solution:
The price per acres is the total price
$200,000 divided by the number of
acres, 14.75.
Solution:
To convert 5.9 hectares to area, we use a unit
fraction with acres in the numerator and
hectares in the denominator.
5.9 ha 1 acre
5.9
5.9 ha =
⋅
=
acres = 14.75 acres
1
0.4 ha
0.4
9/14/2011
Section 9.2
11
price per acre =
9/14/2011
$200,000
≈ $13,559.32/acre
14.75 acres
Section 9.2
12
Measuring Volume
Example - Measuring Volume
9/14/2011
13
Measuring Volume
What is the volume
of this solid in terms
of cubic units??
Solution:
We determine the
volume by counting
the number of cubic
units contained in it.
Here we see how the
cubic units fit within
the region.
There are 18 cubic
units.
14
Volume refers to the amount of space
occupied by a three-dimensional figure.
In order to measure this space, we begin by
selecting a cubic unit, a unit with three sides
(usually labeled length, width, and height),
with each of these sides one unit in length.
Section 9.2
Capacity
1 cubic yard (yd3) = 27 cubic feet (ft³)
Capacity
: The amount of fluid that a
three-dimensional object can hold.
Capacity is a measurement of volume.
English Units of Capacity
1 cubic yard
1 cubic foot (ft³) = (12 in.)³ =
1728 cubic
inches
(in.³)
Section 9.2
9/14/2011
1 cup (c)
= 8 ounces (oz)
2 cups
= 1 pint (pt)
2 pints
= 1 quart (qt)
4 quarts
= 1 gallon (gal)
1 gallon
= 128 ounces
9/14/2011
15
Section 9.2
16
Example -Volume and Capacity in
the English System
Capacity
A swimming pool has a volume of 27,000
cubic feet. How many gallons of water
does the pool hold?
Solution:
We will use dimensional analysis; the unit
fraction is:
Some Volume – Capacity equivalents
Volume in Cubic Units
Capacity
1 cubic yard
≈ 200 gallons
1 cubic foot
≈ 7.48 gallons
231 cubic inches
≈ 1 gallon
9/14/2011
Section 9.2
7.48 gal
1 ft 3
27,000 ft 3 7.48gal
⋅
=
1
1 ft 3
27,000 ( 7.48 ) gal = 201,960 gal
27,000 ft 3 =
17
9/14/2011
Section 9.2
18
Units of Capacity in the Metric
System
Volume and Capacity in the Metric
System
Symbol
Unit
Meaning
kL
kiloliter
1000 liters
hL
hectoliter
100 liters
daL
dekaliter
10 liters
L
liter
1 liter ≈ 1.06 quarts
dL
deciliter
0.1 liter
cL
centiliter
0.01 liters
mL
milliliter
0.001 liter
9/14/2011
Section 9.2
Volume in Cubic Units
19
Capacity
1 cm³
=
1 mL
1 dm³ = 1000 cm³
=
1L
1 m³
=
1 kL
Section 9.2
9/14/2011
Example - Volume and Capacity in
the Metric System
Example - Volume and Capacity in
the Metric System
An aquarium has a volume of 36,000 cubic
centimeters. How many liters of water does
the aquarium hold?
Solution:
Note that we could also have used
dimensional analysis; the unit fraction is:
To convert in metric, just move the decimal
place. Since there are 1000 cm3 in one
liter, move the decimal place 3 times to
the right:
1L
1000 cm 3
So
36,000 cm3 = 36,000 cm3 ⋅
9/14/2011
Section 9.2
9/14/2011
Section 9.2
1L
36,000
=
L = 36 L
1000 cm3
1000
36,000. cm3 = 36 L
Section 9.2
21
22