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Section 11.1 - Linear Measure
The English System
Originally, a yard was the distance from the tip of the nose to the end of an outstetched arm of
an adult person and a foot was the length of a human foot. Since then it has gone through many
definitions until now the definitions are based on the meter.
Unit
Equivalent in Other Units
yard (yd)
3 feet
foot (ft)
12 inches
mile (mi)
1760 yards or 5280 feet
Dimensional Analysis (Unit Analysis)
A process used to convert from one unit of measure to another using unit ratios (ratios equivalent
to 1).
Example 1: Convert each of the following:
a) 200 feet =
b) 3.75 yards =
c) 8690 feet =
d) 940 inches =
yards
inches
miles
yards
The Metric System
Unit
kilometer
hectometer
dekameter
meter
decimeter
centimeter
millimeter
Symbol Relationship to Base Unit
km
1000 m
km
100 m
dam
10 m
m
base unit
dm
0.1 m
cm
0.01 m
mm
0.001 m
Approximate Conversions Between English and Metric Systems
• 1 kilometer ≈ 0.62 miles
• 1 meter ≈ 1.09 yards
• 2.54 centimeters ≈ 1 inch
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Example 2: Convert each of the following:
a) 3.5 km =
m
b) 375 cm =
hm
c) 765 mm =
d) 5.8 km =
e) 70 miles/hour =
f) 100 yards =
dm
cm
km/hour
meters
Example 3: 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 bill
d) $100 bill
e) $1000 bill
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Greatest Possible Error:
The greatest possible error (GPE) of a measurement is one-half the smallest unit used.
Example 4: Determine the GPE for each of the following measurements and interpret.
a) 25 inches
b) 10.8 cm
c) 5.64 m
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 points 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.
Distance Around a Plane Figure
The perimeter of a simple closed curve is the length of the curve. If a figure is a polygon, its
perimeter is the sum of the lengths of its sides. A perimeter has linear measure.
Example 5: Find the perimeter of each of the shapes below:
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Example 6: Given a square of any size, stretch a rope tightly around it. Now take the rope off, add 100 inches
to it, and put the extended rope back around the square so that the new rope makes a square around the
original square. Find d, the distance between the squares.
Circumference of a Circle
A circle is defined as the set of all points in a plane that are the same distance from a given point,
the center. The perimeter of a circle is its circumference.
C = 2π r = π d
where r is the radius, d is the diameter, and π = 3.14159....
Example 7: Find the circumference of the circle pictured below:
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Arc Length
The length of an arc depends on the radius of the circle and the central angle determing the arc.
Example 8: Determine the length of each arc described below of a circle with radius r:
a) Semi-circle
b) Quarter Circle
c) Arc with central angle θ .
Example 9: Find the following:
a) the radius of a circle that has a circumference of 18π meters.
b) the length of a 35◦ arc of a circle with radius 15 cm.
c) the radius of a circle that has an arc with central angle 85◦ and length of 150 cm.
Section 11.1 Homework Problems: 1, 3, 6, 11, 13, 17, 19, 23, 25-27, 30-32
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Section 11.2 - Areas of Polygons and Circles
Area is measured using square units and the area of a region is the number of non-overlapping
square units that covers the region. For instance, a square measuring 1 cm on a side has an area of
1 square centimeter denoted 1 cm2 .
Areas on a Geoboard
In teaching the concept of area, intuitive activities should preced the development of formulas.
Example 1: Find the area of each of the figures below:
Converting Units of Area
The most commonly used units of area in the English system are the square inch (in.2 ), the square
foot (ft2 ), the square yard (yd2 ), and the square mile (mi2 ). In the metric system, the most commonly used units are the square millimeter (mm2 ), the square centimeter (cm2 ), the square meter
(m2 ), and the square kilometer (km2 ). We must be careful when coverting between these units.
Example 2: Convert each of the following:
a) 1 m2 =
cm2
b) 9 yd2 =
ft2
c) 5 cm2 =
mm2
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d) 124 km2 =
m2
e) 3000 ft2 =
yd2
f) 14,256in2 =
yd2
Land Measure
One application of area today is in land measures. The common unit of land measure in the English
system is the acre. Historically, an acre was the amount of land a man with one horse could plow
in one day.
Unit of Area
1 acre
1 mi2
1 a (are)
1 ha (hectare)
1 km2
Equivalent in Other Units
4840 yd2
640 acres
100 m2
100a or 10,000m2 or 1hm2
1,000,000 m2
Example 3:
a) A square field has a side of 400 yards. Find the area of the field in acres.
b) A square field has a side of 400 meters. Find the area of the field in hectares.
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Area Formulas
Let’s develop the formulas for the areas of specific shapes.
Rectangle
Parallelogram
Triangle
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Trapezoid
Regular Polygon
Circle
Sector of a Circle
r
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Summary
Figure
Area
Variables
Rectangle
A = lw
l =length, w =width
Parallelogram
A = bh
b =base, h =height
Triangle
A = 12 bh
b =base, h =height
Trapezoid
A = 12 (b1 + b2 )h b1 , b2 =bases, h =height
Regular Polygon
A = n( 21 sa)
n =number of sides, a =apothem, s =length of side
Circle
A = π r2
r =radius
θ◦
(π r 2 )
Sector of a Circle A =
◦
360
θ =angle;r =radius
Example 4: Find the area of each of the figures below:
Section 11.2 Homework Problems: 4-7, 9, 16, 17, 20-22, 25-35, 37, 40, 42
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Section 11.3 - The Pythagorean Theorem and the Distance Formula
Theorem 11-1 (Pythagorean Theorem)
If a right triangle has legs of lengths a and b and hypotenuse of length c, then c2 = a2 + b2 .
Example 1: Proof 1 of the Pythagorean Theorem (hypotenuse outside)
Example 2: Proof 2 of the Pythagorean Theorem (hypotenuse inside)
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Example 3: The size of a rectangular television screen is given as the length of the diagonal of the screen. If
the width of the screen is 24 inches, and the height of the screen is 18 inches, what is the length of the
diagonal?
Example 4: A pole BD, 28 feet high, is perpendicular to the ground. Two wires BC and BA, each 35 feet
long, are attahced to the top of the pole and to stakes A and C on the ground. If points A, D, and C are
collinear, how far are the stakes A and C from each other?
Special Right Triangles
Property of 45◦ − 45◦ − 90◦ triangle:
In an isosceles right triangle, if the length of each leg is a,
√
then the hypotenuse has length a 2.
Property of 30◦ − 60◦ − 90◦ triangle: In a 30◦ − 60◦ − 90◦ triangle, the length of the hypotenuse
is
√
◦
◦
two times as long as the leg opposite the 30 angle and the leg opposite the 60 angle is 3 times
the shorter leg.
Example 5: Find x and y in the following figures:
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Theorem 11-2 (Converse of the Pythagorean Theorem) If △ABC is a triangle with sides of lengths a, b,
and c such that a2 + b2 = c2 , then △ABC is a right triangle with the right angle opposite the side of
length c.
Example 6: Determine if the following can be the lengths of the sides of a right triangle:
a) 51, 68, 85
b) 3, 4, 7
Distance Formula
The distance between the points A(x1 , y1 ) and B(x2 , y2 ) is given by
q
AB = (x2 − x1 )2 + (y2 − y1 )2
Example 7: Let’s convince ourselves of the distance formula.
Example 8: Find the distance between the points (−5, −3) and (2, −1)
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Example 9: Show that (7, 4), (−2, 1), and (10, −5) are the vertices of an isosceles triangle.
Example 10: Show that (0, 6), (−3, 0), and (9, −6) are the vertices of a right triangle.
Example 11: Determine whether or not the points (0, 5), (1, 2), and (2, −1) are collinear.
Section 11.3 Homework Problems: 1, 2, 8, 9, 11, 12, 14-17, 27-30, 36-41, 53-55
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Section 11.4 - Surface Areas
The surface area is the sum of the areas of the faces (lateral and bases) of a three-dimensional
object. The lateral surface area is the sum of the areas of the lateral faces.
Surface Area of a Cube
s
s
s
The surface area of a cube is
, where s is the length of a side.
Example 1: Find the surface area of a cube with length of a side 8 inches.
Surface Area of a Right Prism with Regular n-gon Bases
h
The surface area of a Right Prism with regular n-gon bases is
, where l is the length
of a side of the base, h is the height of a lateral face, and B is the area of the base.
Example 2: Find the surface area of a right regular-hexagonal prism with height 7 feet and length of each
side of the hexagon 4 feet.
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Surface Area of a Right Regular Cylinder
h
r
The surface area of a right circular cylinder is
and h is the height of the cylinder.
, where r is the radius of the circle,
Example 3: Find the surface area of a right circular cylinder in which the radius of the circular base is 5 cm
and the height of the cylinder is 25 cm.
Surface Area of a Right Regular Pyramid
l
b
The surface area of a right regular pyramid is
, where n is the number of sides of
the regular polygon, b is the length of a side of the base, B is the area of the base, and l is the slant
height.
Example 4: Find the surface area of a right regular triangular pyramid with slant height 5 inches and length
of side of the base 4 inches.
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Surface Area of a Right Circular Cone
l
h
r
The surface area of a right circular cone is
l is the slant height.
, where r is the radius of the circle, and
Example 5: Find the surface area of a right circular cone with height 4 cm and radius 3 cm.
Surface Area of a Sphere
r
The surface area of a sphere is
‘
, where r is the radius of the sphere.
Example 6: Find the surface area of a sphere with diameter 16 inches.
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Example 7: The napkin ring pictured in the following figure is to be resilvered. How many square millimeters must be covered?
Example 8: The base of a right pyramid is a regular hexagon with sides of length 12 meters. The altitude of
the pyramid is 9 meters. Find the total surface area of the pyramid.
Section 11.4 Homework Problems: 1-15, 26, 38-40
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Section 11.5 - Volume, Mass, and Temperature
Surface Area is the number of square units covering a three dimensional figure; Volume describes
how much space a three-dimensional figure contains.
The unit of measure for volume must be a shape that tessellates space (can be stacked so that they
leave no gaps and fill space). Standard units of volume are based on cubes and are cubic units. A
cubic unit is the amount of space enclosed within a cube that measures 1 unit on a side.
Example 1: Determine the surface area and volume of the following figure:
Volume of Right Rectangular Prisms
The volume of a right rectangular prism can be measured by determining how many cubes are
needed to build it as a solid.
Thus, the volume of a right rectangular prism is
base, w is the width of the base, and h is the height of the prism.
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Converting Metric Measures of Volume
The most commonly used metric units of volume are the cubic centimeter and the cubic meter.
Example 2: Convert each of the following:
a) 5m3 =
cm3
m3
b) 12,300cm3 =
In the metric system, cubic units may be used for either dry or liquid measure, although units such
as liters and milliliters are usually used for liquid measures. By definition, a liter, symbolized L,
equals the capacity of a cubic decimeter (1L= 1dm3 )
Example 3: Convert each of the following:
a) 3cm3 =
L
b) 1mL=
cm3
c) 4.2kL=
m3
d) 68L=
mL
e) 9m3 =
L
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Converting English Measures of Volume
Basic units of volume in the English system are the cubic foot (1ft3 ), the cubic yard (1yd3 ), and
the cubic inch (1in3 ). For liquid measures we use the gallon and the quart.
1 gallon = 231 in3
1 quart = 14 gallon
Example 4: Convert each of the following:
a) 45yd3 =
ft3
yd3
b) 4320in3 =
c) 3ft3 =
yd3
d) 1.3ft3 =
gallons
d) 30ft3 =
quarts
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Volumes of Specific Figures:
Right Prism
h
The volume of a Right Prism is
of the base.
, where h is the height of the prism and B is the area
Example 5: Find the volume of a right regular-hexagonal prism with height 7 feet and length of each side of
the hexagon 4 feet.
Right Circular Cylinder
h
r
The volume of a right circular cylinder is
h is the height of the cylinder.
, where r is the radius of the circle, and
Example 6: Find the volume of a right circular cylinder in which the radius of the circular base is 5 cm and
the height of the cylinder is 25 cm.
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Right Pyramid
l
b
The volume of a right pyramid is
the area of the base.
, where h is the height of the pyramid and B is
Example 7: Find the volume of a right regular triangular pyramid with height 5 inches and length of side of
the base 4 inches.
Right Circular Cone
l
h
r
The volume of a right circular cone is
the height of the cone.
, where r is the radius of the circle, and h is
Example 8: Find the volume of a right circular cone with height 4 cm and radius 3 cm.
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Sphere
r
The volume of a sphere is
‘
, where r is the radius of the sphere.
Example 9: Find the volume of a sphere with diameter 16 inches.
Mass
Mass is a quantity of matter. Weight is a force exerted by gravitational pull. On Earth, the terms
are commonly interchanged.
In the metric system, the fundamental unit for mass is the gram, denoted g. A paper clip and a
thumbtack each have a mass of about 1 gram. One mL of water weighs about 1 gram.
Example 10: How many liters of water can a 90 cm by 160 cm by 65 cm rectangular prism hold? What is
the mass in kilograms?
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Temperature
To measure temperature in the metric system the degree Celsius is used. To measure temperature in
the English System, the Fahrenheit scale is used. These two scales have the following relationship:
Example 11: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of
Celsius temperature. Use it to convert 65◦ C to ◦ F.
Example 12: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of
Fahrenheit. Use it to convert 100◦ F to ◦ C.
Section 11.5 Homework Problems: 1-7, 9-16, 20, 22, 24, 25, 34, 35, 43, 44, 46-48, 63-66
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