Section 13-1 – Linear Measure
Units of Length in the English System:
1 yard = 3 feet = 36 inches
1 foot = 12 inches
1 mile = 1760 yards = 5280 feet
1 foot = 1/3 yard = 1/5280 mile
1 inch = 1/12 foot = 1/36 yard = 1/63360 mile
1 yard = 1/1760 mile
We can convert from one unit of measure to another using “dimensional analysis”.
Example: If a car is traveling at 65 miles per hour, what is its speed in feet per second?
Example: Convert 5432 yards per minute into miles per hour
Example: Convert 64 inches into yards
Example: Convert 0.4 miles into feet.
Example: Convert 0.875 feet into yards.
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The Metric System
Unit
Kilometer
Hectometer
Dekameter
Meter
Decimeter
Centimeter
Millimeter
Conversion Relationship
Kilo
103
hecto
102
Symbol
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
(see pg. 843)
deka
101
base (unit)___
100
deci
10-1
centi
10-2
milli
10-3
Example: Convert the following
a) 1.7 km into meters
b) 385 mm into meters
c) 0.08 km into centimeters
d) 15 mm into centimeters
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Distance Properties:
1. The distance between any two points A and B is greater than or equal to 0. (AB ≥ 0)
2. The distance between any two points A and B is the same as the distance between points B
and A. (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 (AB + BC ≥ AC)
If A, B, and C are collinear and B is in between A and C, then AB + BC = AC.
A
B
C
If A, B, and C are NOT collinear (and form a triangle), then AB + BC > AC.
This is known as the Triangle Inequality.
B
A
C
Example: Two sides of a triangle are 31 cm and 85 cm long and the measure of the third side must
be measured in centimeters.
a) What is the longest the third side can be?
b) What is the shortest the third side can be?
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Perimeter and Circumference
The perimeter of a simple closed curve is the length of the curve.
Example: Find the perimeter of the following:
a) A square with 5 mm sides.
b) A rectangle with length 8 feet and width 3 feet.
The perimeter of a circle is called circumference.
In the late 18th century, mathematicians proved that the ratio of circumference to diameter is pi.
ie,
C

d
So, C = _______________ or C = __________________
Example: Find the circumference of a circle with :
a) A diameter of 12 inches
b) A radius of 0.5 cm.
Example: If the circumference of a circle is 20π ft, what is the radius of the circle?
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Arc Length: The length of an arc on a circle depends on the radius and the central angle.
Since a circle contains 360°, then 1° is
1
15
of a circle; 15° is
of a circle; etc.
360
360
The length of an arc whose central angle is Ө° determines

360
of a circle.
The central angle in a semi-circle is _________, therefore the arc length of a semi-circle
is ______________________________.
The central angle in a quarter-circle is _________, therefore the arc length of a quarter-circle
is ______________________________.
Therefore, an arc of Ө° will have length ___________________________________.
Example: Calculate the length of an arc with:
a) Central angle 45° and a radius of 10 mm.
b) Central angle 124° and a radius of 5 feet.
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Section 13-2 – Areas of Polygons and Circles
Finding Areas on a Geoboard
Examples: Find the area of the following figures
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Areas of Polygons:
Area of a Rectangle:
A  lw
Area of a Parallelogram:
A  __________
Area of a Triangle:
A  _____________
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Area of a Kite:
A  _______________
Area of a Trapezoid:
A  _______________
Area of a Regular Polygon:
A  _____________________
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Area of a Circle:
A  _________________________
Area of a Sector of a Circle:
A  ___________________________
Examples: Find the area of each of the following:
a)
2 ft
6 ft
b)
10 mm
15 mm
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c)
8 yd
4 yd
11 yd
d)
1.5 cm
2.2 cm
e)
3 ft
2 ft
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f)
1.05 m
g)
28.2 in
h)
8 in
8 in
20 inches
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Example: Complete the table and answer the questions that follow.
Starting Dimensions
Length
Width
3 ft
2 ft
4 in
3 in
10 yd
8 yd
Perimeter
Area
Multiply length and width by 2
Starting Dimensions
Multiply length and width by 2
Starting Dimensions
Multiply length and width by 2
a) When you double the length and width of the rectangle, does the perimeter double?
b) When you double the length and width of the rectangle, does the area double?
b) What happens to the area of a rectangle when you double the length and width?
Example: If the ratio of the sides of two squares is 1 to 5, what is the ratio of their areas?
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Converting Units of Area (metric):
1m
100 cm
1m
1m
1000mm
100 cm
1000 mm
0.001 km
1m
0.001 km
Example: Complete the following conversions:
a) 650 cm2 into m2
b) 650 cm2 into mm2
c) 35 km2 into m2
d) 90,000,000 mm2 into km2
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Converting Units of Area (English):
1 ft
1 yd
1 mi
(1760 yd)
(5280 ft)
Unit of Area
Equivalent in other units
1
1 2
or
yd
or
mi2
9
27,878,400
1
or 1296 in2
or
mi2
3,097,600
1 ft2
144 in2
1 yd2
9 ft2
1 mi2
3,097,600 yd2 or 27,878,400 ft2
Example: Complete the following conversions:
a) 5000 ft2 into yd2
b) 150 yd2 into ft2
c) 10,000 yd2 into mi2
d) 25 mi2 into ft2
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Land Measurement:
1 acre = 4840 yd2
1 mi2 = 640 acres
Examples: Convert the following:
a) 1 acre into ft2
b) 11,520 acres into mi2
c) 15 mi2 into acres
Section 13-3 – The Pythagorean Theorem, Distance Formula, and Equation of a Circle
The Pythagorean Theorem: If a right triangle has legs of lengths a and b and hypotenuse of
length c, then a2 + b2 = c2.
Examples: Find the missing side:
a)
x
8
5
14
b)
16
x
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Example: Could the following sides be the sides of a right triangle?
a) 2, 3,
13
b) 3, 4, 7
Example: Two cars depart from the same house at 5:00 pm. One drives south at 50 mph and the
other drives east at 60 mph. At 8:00 pm, how far apart are the two cars?
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The Distance Formula:
The Distance between two points
 x1 , y1  and  x2 , y2  is:
Example: Find the distance between the points
 2,1 and  5,4 
The Equation of a Circle with Center at the Origin:
The Equation of a Circle with Center at (h, k):
________________________________
____________________________________
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Examples:
a) Write the equation of a circle centered at (6, –4) with a radius of 7.
b) Determine the center and radius of the following circle: (x – 3)2 + (y + 4)2 = 25
Section 13-4 – Surface Area
The surface area of a 3 dimensional figure is the sum of the areas of the lateral faces.
Consider a cube and its net:
The Surface Area of a Cube is:
SA = ______________________________________
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Consider a rectangular prism and its net:
The Surface Area of a Rectangular Prism is:
SA = _________________________________
Consider a pentagonal prism and its net:
The Surface Area of a Pentagonal Prism is:
SA = _________________________________
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Consider a cylinder and its net:
The Surface Area of a Cylinder is:
SA = _________________________________
Consider the following right square pyramid and its net:
The Surface Area of a Pyramid is:
SA = _________________________________
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Consider a cone:
The Surface Area of a Cone is:
SA = _________________________________
Consider a sphere:
The Surface Area of a Sphere is:
SA = _________________________________
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Examples: Find the surface area of the following figures:
a)
4 in
2 in
20 in
b)
3 cm
5 cm
c)
1.5 ft
0.75 ft
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d) The following snow-cone
2 in
5 in
e)
6 cm
4 cm
4 cm
e)
A sphere with diameter of 20 in.
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Example: How much material is needed to make the following tent?
3 ft
6 ft
5 ft
Example: How does the surface area of a box change if each dimension is doubled?
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Examples: How does the surface area of a right circular cone change if you triple the radius and
triple the slant height?
Section 13-5 – Volume, Mass, and Temperature
Converting Measures of Volume: (most common are cm3 and m3)
Each metric unit of length is 10 times as great as the next smaller unit.
Each metric unit of area is 100 times as great as the next smaller unit.
Each metric unit of volume is 1,000 times as great as the next smaller unit.
1 m3 = 1,000,000 cm3
(to go from m3 to cm3, move decimal 6 places right)
1 cm3 = 0.000001 m3
(to go from cm3 to m3, move decimal 6 places left)
1 in3 =
1
ft3
1728
1 ft3 =
1
yd3
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1 yd3 = 27 ft3
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Examples: Convert the following:
a) 9 m3 = ____________cm3
b) 13,400 cm3 = _________________ m3
c) 45 yd3 = _____________________ft3
d) 4320 in3 = _______________________ft3 = __________________________yd3
We generally use Liters (mL, L, kL, etc. for liquid measurement)
1 L = 1000 cm3
1 cm3 = 1 mL
a) 27 L = _________________________mL
b) 3 mL = _____________________cm3
c) 5 m3 = _____________________cm3 = _____________________________L
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Volume of a right Rectangular Prism:
V = ____________________
h
w
l
Volumes of all right Prisms and Cylinders:
V = Bh
(B = area of the base)
Examples: Find the volume of the following:
a)
7 cm
b)
2 in
4 in
c)
8 yd
2 yd
5 yd
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Volume of a right Pyramid:
V = _____________________
Volume of a right Cone:
V = _____________________
(see pg. 913)
Examples: Find the volume of the following:
a)
6.5 in
1.5 in
4 in
b)
12 cm
6 cm
Volume of a Sphere:
V = ______________________
Example: Find the volume of a sphere with diameter 8 meters.
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Mass:
The fundamental unit for mass is the gram.
1 kilogram
1 gram
1 milligram
______ g
________g
Examples: Convert the following
a) 64 g = _______________________kg
b) 7524 kg = ________________________g
c) 580 g = _________________________mg
Temperature (Fahrenheit vs. Celsius)
C  95  F  32  95 F  160
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or
F  95 C  32
Examples: Convert the following:
a) 80°F = ________________°C
b) 10°C = ________________°F
c) 212°F = ________________°C
d)
0°C = ________________°F
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Attributes and Units
Measurement is a three-step process: choose an attribute to measure, choose an appropriate unit,
determine how many of these units are necessary to find the length, cover, or fill the object. In this
activity you will focus on the attribute and the unit.
Next to each description are two blanks.
In the first blank, label the attribute that is being measured.
Attribute choices: L (length), A (area), SA (surface area), V (volume)
In the second blank, choose the best unit.
Unit Choices: mm, m, km (length)
in2, ft2, acre (area or surface area)
mL, L, kL (volume)
Problem
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2
3
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7
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Attribute
L
Unit
Km
Description
The distance from New York City to Chicago
The amount of wrapping paper needed to wrap a CD
The height of a 5 story building
The width of a cockroach
The amount of tea in a pitcher
The amount of space covered by a bathroom floor
The amount of gas in a car’s full gas tank
The amount of grass in Central Park
The amount of fabric needed to cover a couch cushion
The width of a two car garage
The wingspan of a hummingbird
The amount of wall space covered by a light switch
The amount of liquid held by a baby bottle
The size of a ceiling to be painted
The amount of paint needed to paint a ceiling
The amount of water in a hot tub
The thickness of an iPhone
The size of a living room rug
The size of a label on a soup can
The amount of aluminum foil needed to cover a baked potato
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