MATH 20043
CHAPTER 10
EXAMPLES AND DEFINITIONS
Section 10.1 Systems of Measurement
It can be helpful to have an everyday approximate equivalent to units of measure.
These are called benchmarks. Some benchmarks for customary units:
•1 inch – diameter of a quarter, length of a small paperclip
•2 inches – width of a credit card
•6 inches – length of a dollar bill
•1 yard – height from floor to doorknob
Volume is the amount of space occupied by an object. Capacity is the amount that
can be contained by an object.
•Liquid volume: ounce, cup, pint, quart, gallon, barrel, hogshead
8 oz = 1 cup 2 c = 1 pint
2 pt = 1 quart
4 qt = 1 gallon
63 gal = I hogshead (beer)
•Dry volume: pint, quart, gallon, peck, bushel
8 oz = 1 cup 2 c = 1 pint
2 pt = 1 quart
4 qt = 1 gallon
2 (dry) gallons = 1 peck
4 pk = 1 bushel (bu)
Cooking measures: 1 Tablespoon = 3 teaspoons
16 T = 1 cup
Mass is a measure of the quantity of matter. Weight is a measure of how heavy
something is. Weight is caused by the force of gravity pulling down upon an object.
An object's weight depends on what planet or moon it's on (unlike mass, which is
constant).
•Ounce, pound, stone, hundredweight, ton
16 oz = 1 pound (A pint’s a pound, the world around)
14 lbs = 1 stone
2000 lbs = 1 ton
Ex. A) Using appropriate units – which is the best unit to describe
• the amount of water in a swimming pool?
(a) cup
(b) pint
(c) quart
(d) gallon
• a bicycle's length?
(a) inches
(b) feet
(c) miles
(d) yards
• weight of a whale?
(a) ounces
(b) pounds
(c) tons
(d) too heavy to weigh
• weight of a pencil?
(a) ounces
(b) pounds
(c) tons
(d) too light to weigh
(c) gallons
(d) ounces
• can of soda (liquid measure)?
(a) pints
(b) quarts
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CUSTOMARY/METRIC CONVERSIONS
KNOW:
1 inch ≈ 2.54 centimeters
1 quart ≈ 1 Liter
1 mile ≈ 1.6 kilometers
2.2 pounds ≈ 1 kilogram
A few mnemonics for metric prefixes:
kilometer hectometer dekameter
meter
decimeter centimeter millimeter
King
Henry
Danced
Merrily
Down
Center
Main
King
Henry
Died
Monday
Drinking
Chocolate
Milk
kilo
Kathy
hecto
Hall
deka
meter/liter/gram
Drinks
Milk/Lemonade/Gatorade
deci
During
centi
Class
milli
Mondays
Incidental Note: The abbreviation ‘K’ is now commonly used for 1000 of something
– dollars, meters, or bytes, for example.
Some metric benchmarks:
Length:
•millimeter – thickness of a dime
•centimeter – width of index fingernail
•meter – doorknob to floor
•kilometer – 9 football fields
Volume: 1 cubic centimeter = 1 milliliter
1 teaspoon ≈ 5 milliliter
1000 mL = 1 L
Weight/mass: 1 cubic centimeter of water masses 1 gram.
1 gram ≈ a paperclip
5 g ≈ a nickel
Ex. B) Use dimensional analysis to convert each of the following:
(a) 6 mm =
cm
(b) 125 cm =
m
(c) 5 inches =
cm
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Ex. C) Choose an appropriate unit of measure for each measurement. Use
customary units, metric units, and (if possible) nonstandard units.
(a) distance from home to a grocery store
(b) weight of a refrigerator
(c) water in a fish tank
(d) height of a door
(e) perfume in a bottle of perfume
Temperature:
C° =
5
( F° ! 32)
9
F° = 32° +
9
C°
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Ex. D) My Celsius thermometer says that I have a fever of 40°C. Should I be
alarmed?
!
•Celsius temperature jingle:
30 is hot,
20 is nice,
10 is chilly,
0 is ice!
***Always remember to use units of measure!
Practice Problems for Section 10.1
1. How might you answer these questions from your students: what measurement
tool would you use to find the
(a) mass of a worm?
(b) length of a bean?
(c) circumference of your wrist?
(d) volume of an apple?
2. Perform the following operations. Express your answer in centimeters.
4.2 m + 53 cm – 2846 mm
3. A third grader measures her height in both centimeters and inches. Which
measurement will have more units? Why?
4. It is a cold day. Which would be a greater increase in temperature, an increase of
10°F or an increase of 10°C? Why?
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Section 10.2 Perimeter and Area
The perimeter of a simple closed curve is the distance around the curve.
The circumference of a circle is its perimeter. The ratio of the circumference of a
circle to its diameter is always pi.
C = 2π r
Ex. E) Determine the circumference of a circle with a diameter measuring 5.8
centimeters. Use 3.1416 as an approximation for pi.
Area is the measure of a region – i.e. the measure of the interior of a closed curve.
Ex. F) True/False – counterexample:
(a) Two figures that are congruent will have the same perimeter.
(b) Two figures that have the same perimeter are congruent.
(c) Two figures that are congruent will have the same area.
(d) Two figures that have the same area are congruent.
Areas of Circles, Triangles, and Quadrilaterals
The altitude or height of a triangle is the perpendicular segment from any vertex to
the line that contains the opposite side.
Area Formulas
1. The area A of a circle that has radius r:
2
A = πr .
2. The area A of a rectangle that has length l and width w:
A = lw.
3. The area A of a triangle that has base b and height h:
A = ½bh.
4. The area A of a parallelogram that has base b and height h:
A = bh.
5. The area A of a trapezoid that has parallel sides of lengths b and c and
height h:
A = ½( b + c)h. The quantity ½( b + c) is referred to as the
average base.
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Ex. G) Find the area of a trapezoid with a base lengths of 5 inches and 9 inches and
a height of 4 inches.
The Pythagorean Theorem
If a right triangle has legs of lengths a and b and a hypotenuse of length
c, then c2 = a2 + b2.
Ex. H) A triangle has side lengths 8, 8, and 10 inches. Find the area of the triangle
to the nearest tenth of an inch.
Ex. I) The radius of the larger circle below is 12 units. The intersection point of the
two smaller circles is the center of the larger circle. (a) Find the combined area of
the two smaller circles in the figure below. (b) Find that part of the area of the outer
circle not included within the two smaller circles. Give all answers as exact values
(in other words, in terms of π).
Section 10.2 Bonus Problem (2 points)
A square has an area of 16 square meters. The square has a circle inscribed within
it, and another circle circumscribing it. Find the areas of these two circles. Give your
answer to the nearest hundredth of a meter.
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