Section 11.1 – Linear Measure
English or Customary Units
Originally, a yard was the distance from the tip of the nose to the end of an outstretched arm of an adult
person and a foot was the length of a human foot. Since then, the definition has changed numerous
times and is now based in international standards.
Unit
yard (yd)
foot (ft)
mile (mi)
Equivalent in Other Units
3 ft
12 in.
1760 yd, or 5280 ft
Dimensional Analysis (Unit Analysis)
Dimensional analysis is a process of converting from one unit of measure to another using unit
ratios (ratios equivalent to 1).
Example: Convert the following:
200 ft = _______ yd
3.75 yd = _______ in
940 in = _______ yd
8690 ft = _______ mi
The Metric System
The metric system is a standard based on the meter. It is the official system of measurement
everywhere except Myanmar, Liberia, and the United States.
Unit
kilometer
hectometer
dekameter
meter
decimeter
centimeter
millimeter
Abbreviation
km
hm
dam
m
dm
cm
mm
Relationship to base unit
1000 m
100 m
10 m
base unit
0.1 m
0.01 m
0.001 m
Approximate Conversions Between the English and Metric Systems
• 1 km ≈ 0.62 mi
• 1 m ≈ 1.09 yd
• 2.54 cm ≈ 1 in
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Example: Convert the following:
3.5 km = _______ m
375 cm = _______ hm
765 mm = _______ dm
5.8 km = _______ cm
70 miles/hour = _______ km/hr
100 yds = _______ m
Greatest Possible Error
The greatest possible error (GPE) of a measurement is one-half the smallest unit used.
Example: Determine the GPE for each of the following measurements and interpret.
a. The height of a window is measured as 60 in,
b. The width of a piece of board is measured as 10.8 cm.
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.
a. Triangle Inequality The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
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.
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Example: What is the perimeter of the figure?
75 mm
4 cm
3 cm
60 mm
5 cm
Example: Given a square of any size, stretch a rope tightly around it. Now take the rope off, add
125 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 the set of all points in a plane that are equidistant from a given point, the
center. The distance is the radius.
• The circumference of a circle is the perimeter and is found by C = 2πr = πd, where r is
the radius, d is the diameter, and π = 3.14159...
Arc Length
The arc length is a linear distance around a portion of the circumference of a circle. The length
of an arc depends on the radius of the circle and the central angle determining the arc.
Example: Determine the length of each arc described below for a circle with radius r.
a. Semicircle
b. Quarter circle
c. Arc with central angle θ
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Example: Find the following:
a. The circumference of a circle of diameter 3 meters.
b. The radius of a circle of circumference 18π meters.
c. The length of a 35° arc of a circle of radius 15 cm.
d. The radius of an arc whose central angle is 85° and whose arc length is 150 cm.
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Section 11.2 – Areas of Polygons and Circles
Area is measured using square units. The area of a region is the number of nonoverlapping
square units that cover the region. For instance, a square measuring 1 cm on a side has an area of1
square centimeter denoted 1 cm2.
Areas on a Geoboard
In teaching the concept of area, intuitive activities should precede the development of formulas.
Many such activities can be accomplished using a geoboard or dot paper.
Example: Find the area of each figure 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).
Unit
Square kilometer
Square hectometer
Square decameter
Square meter
Square decimeter
Square centimeter
Square millimeter
Abbreviation Relationship to Square Meter
km2
1,000,000 m2
hm2
10,000 m2
2
dam
100 m2
m2
1 m2
dm2
0.01 m2
2
cm
0.0001 m2
mm2
0.000001 m2
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Example: Convert the following:
1 m2 = _______cm2
9 yd2 = _______ft2
8 cm2 = _______mm2
124 km2 = _______m2
3,000 ft2 = _______yd2
14,256 in2 = _______ 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 a (are)
1 ha (hectare)
1 km2
1 acre
1 mi2
Equivalent
100 m2
100 a = 10,000 m2 = 1 hm2
1,000,000 m2
4840 yd2
640 acres
Example:
a. A square field has a side of 400 yards. Find the area of the field in acres.
b. Find the area of a 3 km by 5 km rectangular 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
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Figure
Rectangle
Parallelogram
Triangle
Trapezoid
Regular Polygon
Circle
Area
A = lw
A = bh
A = ½ bh
A = ½ (b1+b2)h
A = n( ½ sa)
A = πr2
Sector of a circle
A=
!°
(πr2)
360°
variables
l = length; w = width
b = base; h = height
b = base; h = height
b1, b2 = bases; h = height
n = # sides; a = apothem; s = length of side
r = radius
θ = angle; r = radius
Example: Find the area of the regular polygon.
Example: Find the area of the smaller sector.
6 cm
80°
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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: Proof 1 of the Pythagorean Theorem.
Example: Proof 2 of the Pythagorean Theorem.
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Example: The size of a rectangular television screen is given as the length of the diagonal of the
screen. If the length of the screen is 24 in and the width is 18 in, what is the diagonal length?
What is the size of the viewing area? What is the size of the viewing area of a square screen with
the same diagonal length?
Example: A pole BD , 28 feet high, is perpendicular to the ground. Two wires BC and BA ,
each 35 feet long are attached 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
•
The 45°-45°-90° Triangle In an isosceles right triangle, if the length of each leg is a,
then the hypotenuse has length a 2 .
•
The 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° is
the shorter leg.
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3 times
Converse of the Pythagorean Theorem
If ΔABC 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 angles opposite the side of length c.
Example: Determine if the following can be the lengths of the sides of a right triangle:
a. 51, 68, 85
b. 2,
5,3
The Distance Formula: An Application of the Pythagorean Theorem
Example: Find the distance between the two points.
Distance Formula
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2
The distance between the points A(x1, y1) and B(x2, y2) is given by AB = (x2 ! x1 ) + ( y2 ! y1 )
Example: Show that A(7, 4), B(−2, 1), and C(10, −5) are the vertices of an isosceles triangle.
Example: Show that A(0, 6), B(−3, 0), and C(9, −6) are the vertices of a right triangle.
Example: Determine whether the points (0, 5), (1, 2), and (2, −1) are collinear.
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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 Right Prisms
The surface area of a cube is ______________, where s is the length of a side (edge).
The surface area of a right prism with regular n-gon bases is ______________, where p is the
perimeter of the base, h is the height of a lateral face, and B is the area of the base.
Example: Find the surface area of a right regular-hexagonal prism with height 5 feet and length
of each side of the hexagon 4 feet.
Surface Area of a Cylinder
As the number of sides of a right regular prism increases, the figure approaches the shape of a
right regular cylinder.
The surface area of a right circular cylinder is ______________, where r is the radius of the
circle, and h is the height of the cylinder.
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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 Pyramid
The surface area of a right regular pyramid is ______________, where n is the number of faces, l
is the slant height, b is the length of a side of the base, and B is the area of the base.
Example: Find the surface area of a right regular triangular pyramid with slant height 5 in. and
length of side of the base 4 in.
Surface Area of a Cone
The surface area for a right circular cone is ______________, where r is the radius of the circle,
and l is the slant height.
Example: Find the surface area of the cone with height 4 cm and radius 3 cm.
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Surface Area of a Sphere
The surface area of a sphere is _______________, where r is the radius of the sphere.
Example: Find the surface area of a sphere with diameter 16 inches.
Example: 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.
Example: A machine part is in the shape of a hollowed right cylinder. If R1 is 6mm, R2 is 4mm,
and the part is 14mm long, what is the surface area?
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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. 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: Determine the surface area and volume of the following figure made up of unit cubes.
Assume that any hidden cubes are present.
Volume of Right Rectangular Prisms
The volume of a right rectangular prism can be measured by determine how many cubes are
needed to build it as a solid.
The volume of a right rectangular prism is ___________ where l is the length of the 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: Convert each of the following:
a. 5 m3 = _________ cm3
b. 12,300 cm3 = ________ m3
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. A liter (L) is defined as a cubic
decimeter (1 L = 1 dm3).
Example: Convert each of the following:
a. 3 cm3 = _________ L
b. 1mL = _________ cm3
c. 4.2kL = _________ m3
d. 68L = _________ mL
Converting English Measures of Volume
Basic units of volume in the English system are the cubic foot (1 ft3), the cubic yard (1 yd3), and the
cubic inch (1 in3).
In the United States, 1 gal = 231 in.3 ≈ 3.8 L
1 qt = ¼ gal. ≈ 58 in.3
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Example: Convert each of the following:
a. 45 yd3 = _________ ft3
b. 4320 in3 = _________ yd3
c. 3 ft3 = _________ yd3
d. 1.3 ft3 = _________ gallons
e. 30 ft3 = _________ quarts
Volumes of Prisms, Cylinders, Pyramids, Cones, and Spheres
The volume of a right prism with regular n-gon bases is _____________, where h is the height of
a lateral face, and B is the area of the base.
Example: Find the volume of a right regular-hexagonal prism with height 5 feet and length of
each side of the hexagon 4 feet.
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Volume of a Cylinder
The volume of a right circular cylinder is ______________, where r is the radius of the circle,
and h is the height of the cylinder.
Example 3: 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.
Volume of a Pyramid
The volume of a right regular pyramid is ______________, where h is the height and B is the area
of the base.
Example: Find the volume of a right regular triangular pyramid with height 5 in. and length of
side of the base 4 in.
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Volume of a Cone
The volume for a right circular cone is ______________, where r is the radius of the circle, and
h is the height.
Example: Find the volume of the cone with height 4 cm and radius 3 cm.
Volume of a Sphere
The volume of a sphere is _______________, where r is the radius of the sphere.
Example: Find the volume of a sphere with diameter 16 inches.
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Mass
Mass is the quantity of matter. Weight is the force exerted by gravitational pull. 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 g. 1 L of water has a mass of 1 kg.
Unit
Abbreviation
Ton (metric)
Kilogram
Hectogram
Dekagram
Gram
Decigram
Centigram
Milligram
t
kg
hg
dag
g
dg
cg
Mg
Relationship
to Gram
1,000,000 g
1000 g
100 g
10 g
1g
0.1 g
0.01 g
0.001 g
Example: 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. The Kelvin scale is used for scientific
purposes. These scales have the following relationship:
Freezing point of water
Boiling point of water
Celsius
0°C
100°C
Fahrenheit
32°F
212°F
Kelvin
273 K
373 K
Example: 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: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in
terms of Fahrenheit temperature. Use it to convert 100°F to °C.
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