9.6: Summary of Key Concepts

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Summary of Key Concepts

Measurement
Measurement is comparison to some standard.

Standard Unit of Measure
A quantity that is used for comparison is called a standard unit of measure.

Two Types of Measurement Systems
There are two major types of measurement systems in use today. They are the United States system and the metric system.

Unit Fraction
A unit fraction is a fraction that has a value of 1. Unit fractions can be used to convert from one unit of measure to another.

Meter, Liter, Gram, and associated prefixes
Common units of measure in the metric system are the meter (m), for length, the liter (L), for volume, and the gram (g), for mass. To each of these units, a prefix can be attached.

kilo: thousand
deci: tenth
hecto: hundred
centi: hundredth
deka: ten
milli: thousandth

Metric Conversions
To convert from one metric unit to another:

Determine the location of the original number on the metric scale.
Move the decimal point of the original number in the same direction and the same number of places as is necessary to move to the metric unit you wish to convert to.

Denominate Numbers
Numbers that have units of measure associated with them are denominate numbers. The number 25 mg is a denominate number since the mg unit is associated with the pure number 25. The number 82 is not a denominate number since it has no unit of measure associated with it.

Simplified Denominate Number
A denominate number is simplified when the number of standard units of measure associated with it does not exceed the next higher type of unit. 55 min is simplified, whereas 65 min is not simplified

Addition and Subtraction of Denominate Numbers
Denominate numbers can be added or subtracted by

writing the numbers vertically so that the like units appear in the same column.
adding or subtracting the number parts, carrying along the unit.
simplifying the sum or difference.

Multiplying a Denominate Number by a Whole Number
To multiply a denominate number by a whole number, multiply the number part of each unit by the whole number and affix the unit to the product.

Dividing a Denominate Number by a Whole Number
To divide a denominate number by a whole number, divide the number part of each unit by the whole number beginning with the largest unit. Affix the unit to this quotient. Carry the remainder to the next unit.

Polygon
A polygon is a closed plane (flat) figure whose sides are line segments (portions of straight lines).

Perimeter
The perimeter of a polygon is the distance around the polygon.

Circumference, Diameter, Radius
The circumference of a circle is the distance around the circle. The diameter of a circle is any line segment that passes through the center of the circle and has its endpoints on the circle. The radius of a circle is one half the diameter of the circle.

The number \(\pi\)
The symbol \(\pi\), read "pi," represents the nonterminating, nonrepeating decimal number 3.14159... . For computational purposes, \(\pi\) is often approximated by the number 3.14.

Formula
A formula is a rule for performing a task. In mathematics, a formula is a rule that directs us in computations.

Circumference Formulas ([link])
\(C = \pi \cdot d C \approx (3.14)d\)
\(C = 2 \cdot \pi \cdot r C \approx 2(3.14)r\)

Area
The area of a surface is the amount of square length units contained in the surface.

Volume
The volume of an object is a measure of the amount of cubic length units contained in the object.

Area Formulas
Triangle: \(A = \dfrac{1}{2} \cdot b \cdot h\)
Rectangle: \(A = l \cdot w\)
Parallelogram: \(A = b \cdot h\)
Trapezoid: \(A = \dfrac{1}{2} \cdot (b_1 + b_2) \cdot h\)
Circle: \(A = \pi \cdot r^2\)

Volume Formulas ([link])
Rectangle solid: \(V = l \cdot w \cdot h\)
Sphere: \(V = \dfrac{4}{3} \cdot \pi \cdot r^3\)
Cylinder: \(V = \pi \cdot r^2 \cdot h\)
Cone: \(V = \dfrac{1}{3} \cdot \pi \cdot r^2 \cdot h\)