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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

 

SECTION 25.1: WHAT ARE "UNITS"?

"Units" or "units of measurements" are labels which distinguish one type of measurable quantity from other types. A number always precedes a "unit." I will deal with units of "length".

The measurable quantity "length" is determined by a "ruler." See below:


Figure

EXAMPLES OF THE DIFFERENT UNITS OF LENGTH
USED IN SENTENCES
SENTENCE   UNIT OF LENGTH USED
John walked 10 miles today   miles
Mary is 5 feet tall.   feet
Tom bought 6 yards of carpeting for his room.   yards
The road from Bangor to Nova Scotia is 540 kilometers long.   kilometers

There are numerous units of length: centimeter, millimeter, Angstrom, inch, etc. Also, there are numerous units having nothing to do with length, e.g. pounds, degrees centigrade, grams, dollars, etc.

 

SECTION 25.2 WHAT IS A "PLANE FIGURE"?

I will be speaking in everyday language. The language that I'll use is not considered precise mathematical language.

Let's consider the concept of a "plane." A "plane" is a "flat surface." A piece of paper can be thought of as a plane. The top of a table can be thought of as a plane. An "object" drawn on a "plane" is called a "plane figure."


Figure

SECTION 25.3: THE MEANING OF AREA

I am assuming you have a basic understanding of a "square". A "square" is a 4 sided plane figure, i.e. a quadrilateral, with all sides equal in length and a 90 degree (90°) angle where the sides intersect.

1

Consider the following squares:


Figure

Figure

Figure

Figure

SECTION 25.4: EXAMPLES OF AREAS AND THEIR MEANINGS

 

MEANING OF AREA Z units2

Z units2

where Z is a real number such that Z ≥ 0

means

The amount of surface on a plane made up of
a) Z squares
and
b) the sides of each square has a length of 1 unit.

Comment: The unit could be inches, feet, yards, miles, etc. In effect the unit can be any measurement of length.

 

Consider the following examples of areas:

ex. 1) 10 in.2

Pronounced 10 square inches.

Meaning: A surface that contains 10 squares, each square having a side of 1 inch.

Visual representation: Figure

Comment: I could rearrange the above squares in any way I want and I would still have 10 in.2. I could cut the squares up into thousands of bits and pieces and still have 10 in2.

ex. 2) 20 ft.$^{2}$

Pronounced 20 square feet.

Meaning: A surface that contains 20 squares, each square having a side of 1 foot.

Visual representation: Figure

 

ex. 3) The earth has a surface area of 196,940,400 mi.$^{2}$

Pronounced one hundred and ninety six million, nine hundred and forty thousand, four hundred square miles.

Meaning: A surface that contains 196,940,400 squares, each square having a side of 1 mile.

Visual representation: Sorry! Too many squares!

 

ex. 4) The moon has a surface area of 23,000,000 mi.$^{2}$

Pronounced twenty three million square miles.

Meaning: A surface that contains 23,000,000 squares, each square having a side of 1 mile.

Visual representation: Sorry! Too many squares!

 

I need to review a few ideas,

SECTION 25.5: WHAT ARE PARALLEL LINES?

Two lines are "parallel" if they are equally distant from each other.


Figure

 

SECTION 25.6: WHAT ARE PERPENDICULAR LINES?

Two straight lines are "perpendicular" if the angle at their point of intersection is 90 degrees(90°).


Figure

 

SECTION 25.7: HASH MARKS AND ARROWS

Figure



HASH MARKS
Hash marks small line segments and are indicators of sides of plane figures having equal lengths. If the sides of a plane figure are marked with the same number of hash marks then those sides are equal.

ex. 1)

Figure

- Sides bc and ad are understood to be equal in length because they have the same number of hash marks: 2 on each side.

- Sides ab and dc are understood to be equal in length because they have the same number of hash marks: 1 on each side.

 

ex. 2)

Figure

- Sides bc, cd, da, and ab are understood to be equal in length because they have the same number of hash marks: 1 on each side.

 

ex. 3)
Figure

- sides ab and ac are understood to be equal in length because they have the same number of hash marks: 3 on each side.

 

SECTION 25.8: THE ARROW MARK FOR PARALLEL LINES

I am assuming that you understand that two lines are parallel when they are equally distant from each other.

Figure   Figure

An "arrow mark" is a small arrow placed along a pair line segments of a plane figure and is used to mark that sides of a plane figure that are parallel. An arrow mark may have more than one arrowhead.

Figure   Figure   Figure

ex. 1)

Figure

- bh is parallel to yj since the sides have the same number of arrow marks: 1 arrow mark on each side.

ex. 2

Figure

- zh is parallel to kv since the sides have the same number of arrow marks: 2 arrow marks on each side.

ex. 3)

Figure

- tn is parallel to we since the sides have the same number of arrow marks: 3 arrow marks on each side.

- wt is parallel to en since the sides have the same number of arrow marks: 2 arrow marks on each side.

 

SECTION 25.9: SOME PLANE FIGURES AND THEIR AREA FORMULAS

Below are some plane figures, their characteristics and their area formulas.

Figure

Figure

Figure

Figure

Figure

SECTION 25.10: PERIMETER OF A PLANE FIGURE


Figure


Figure

EXAMPLES: FIND THE AREA AND PERIMETER OF THE FOLLOWING PLANE FIGURES

 

ex. 1)

Figure

 

ex. 2)

Figure

 

ex. 3)

Figure

 

ex. 4)

Figure

 

ex. 5)

Figure

ex. 6)

3

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

-----------------------------

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES