Book citra math 3_e

CUBE

A. Properties of Cube

a cube is called Hexahedron because It is composed of six square faces

that meet each other at right angles.

It has composed of six shape and each side, there are ABCD, EFGH, ADEH,

BCGF, DCGH and ABFE.

A cube has 12 edges there are ;

CG, FG. Each edge is the same size.

It has eight verticles and make right angles there are point A, B, C, D, E,

F, G, H.

There are Space Diagonal DBHF, CDEF,ACGE, EHBC

B. There are total of 11 distinct

A B

CD

EF

GH

English For Math© Citra Nur Fadzri Yati/1001125036/Math 3E

CUBE (Hexahedron)


that meet each other at right angles.


BCGF, DCGH and ABFE.

A cube has 12 edges there are ; AB, BC, CD, AD, AE, BF, EF, EH, DH, HG,

CG, FG. Each edge is the same size.


Diagonal DBHF, CDEF,ACGE, EHBC

There are total of 11 distinct nets for the cube

English For MathCitra Nur Fadzri Yati/1001125036/Math 3E

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AB, BC, CD, AD, AE, BF, EF, EH, DH, HG,



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C. Surface Area of cube

We will need to calculate the surface area with edge length a is :

Let’s find the surface area of this cube :

a = 5cm

S = 6a² {S equivalent six a square}

= 6x(5 cm)² {S equivalent five centimeter square}

= 6x25 cm² {S equivalent six times twenty five centimeter square}

= 150 cm² {S equivalent is one hundred fifty centimeter square

D. Volume of Cube

The formula to find volume of cube with edge length a is :

Let’s find the Volume of this cube !

a = 10cm

V = a³ {volume is a cubic}

V = (10x10x10) cm³ {Volume is ten times ten times ten centimeter cubic}

V = 1000 cm³ {Volume is one thousand centimeter cubic}

V = a³ {volume is a cubic}

S = 6a2{S = six a square}


3

Cartesian Coordinates

A. Definition of "Cartesian" ... ?

Cartesian coordinates can be used to pinpoint where you are on a map or graph.

They are called Cartesian because the idea was developed by the mathematician and

philosopher Rene Descartes who was also known as Cartesius. He is also famous for saying

"I think, therefore I am".

Using Cartesian Coordinates you mark a point on a graph by how far along and how far up

it is:

The point (12 5{twelve, five}) is 12{twelve} units

along, and 5{five} units up.

B. X and Y Axis

The X Axis runs horizontally through zero

The Y Axis runs vertically through zero

Axis: The reference line from which distances are measured. The plural of Axis is Axes, and

is pronounced ax-eez And you can remember which is which by:

Example:

Point (6,4){six, four} is

6{six} units along (in the x direction), and


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4{four} units up (in the y direction)

So (6,4) {six, four} means: Go along 6 and then go up 4 then "plot the dot".

C. Direction

As x increases, the point moves further right.

When x decreases, the point moves further to the left.

As y increases, the point moves further up.

When y decreases, the point moves further down.

D. Abscissa and Ordinate

You may hear the words "Abscissa" and "Ordinate" ... they are just the x and y values:

Abscissa: the horizontal ("x") value in a pair of coordinates: how far along the point is

Ordinate: the vertical ("y") value in a pair of coordinates: how far up or down the

point is

E. What About Negative Values of X and Y?

Just like with the Number Line, you can also have negative values.

Negative: start at zero and head in the opposite direction:


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So, for a negative number:

go backwards for x

go down for y

For example (-6,4) {six, four} means:

go back along the x axis 6 then go up 4.

And (-6,-4) {six, four} means:

go back along the x axis 6 then go down 4.

F. Four Quadrants

When we include negative values, the x and y axes

divide the space up into 4 pieces:

Quadrants I, II, III and IV

(They are numbered in a counter-clockwise

direction)

In Quadrant I both x and y are positive, but ...

in Quadrant II x is negative (y is still positive),

in Quadrant III both x and y are negative, and

in Quadrant IV x is positive again, while y is negative.


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Like this:

QuadrantX

(horizontal)

Y

(vertical)Example

I Positive Positive(3,2)

{three, two}

II Negative Positive

III Negative Negative

(-2,-1)

{negative two,

negative three}

IV Positive Negative

Note: The word Quadrant comes form quad meaning four. For example, four babies born

at one birth are called quadruplets, a four-legged animal is a quadruped. and a quadrilateral

is a four-sided polygon.

G. Dimensions: 1, 2, 3 {one, two, three}and more ...

Think about this:

1 {first} ⟶ The number line can only go : Left-right

So any position need just one number

2 {second} ⟶ Cartesian coordinates can go : Left-right and Up-down

so any position needs two numbers

3 {third} ⟶ How do we locate a spot in the real world ? We need to know:

⟶ left-right, up-down and forward-backward


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⟶ that is three numbers, or 3 dimensions!

3 {three} Dimensions, In fact, this idea can be continued into four dimensions and more - I

just can't work out how to illustrate that for you! Cartesian coordinates can be used for

locating points in 3{three} dimensions as in this example:

Here the point (2, 4, 5) {two, four, five} is shown in

three-dimensional Cartesian coordinates.


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Percentages (%)

A. Definition

"Percent" comes from the latin Per Centum. The latin word Centum means 100 (a hundred),

for example a Century is 100{a hundred} years.

B. When you say "Percent" you are really saying "per 100 {a hundred}"

1. So 50% {fifty percent} means 50 {fifty} per 100 {a

hundred} 50%{fifty percent} of this box is green

2. And 25% {twenty five percent} means 25{twenty

five} per 100{a hundred}

(25% {twenty five percent} of this box is green)

C. Using Percent

2. Because "Percent" means "per 100" {per a hundred} you should think "this should

always be divided by 100{a hundred}"

3. So 75% {seventy five percent} really means 75/100{Seventy per a hundred}

4. And 100% {a hundred percent is} is 100/100 {a hundred by a hundred}, or exactly 1

(100%{a hundred percent} of any number is just the number, unchanged)

5. And 200% is 200/100, or exactly 2 (200% of any number is twice the number)

D. Percent can also be expressed as a Decimal or a Fraction

E. Examples

A Half can be written...

As a percentage ⟶ 50%

As a decimal ⟶ 0.5

As a fraction ⟶ ½


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1. Calculate 25% of 80 {twenty five percent of eighty}

25% = 25/100 {twenty five percent is twenty five per a hundred}

(25/100) × 80 = 20{twenty five per a hundred times eighty is twenty}

So 25% of 80 is 20 {twenty five of eighty is twenty}

2. 15%{fifteen percent} of 200{two hundred} apples were bad. How many

apples were bad?

15% = 15/100{fifteen percent is fifty per a hundred}

(15/100) × 200 = 15 × 2 = 30 apples{fifteen by a hundred times two hundred is fifteen

times two is thirty}

30{thirty} apples were bad

3. A Skateboard is reduced 25% {twenty five percent} in price in a sale. The

old price was Rp 100.000 {a hundred of thousand rupiah}. Find the new

price

First, find 25% of Rp 100.000 {twenty five percent of a hundred thousand rupiah}:

25% = 25/100 {twenty five percent is twenty five per a hundred}

(25/100) × Rp 100.000 = Rp 25.000 {twenty five per a hundred times a hundred of

thousand rupiah is Twenty five thousand rupiah}

25% of Rp 100.000 is Rp 25.000 {twenty five percent of a hundred of thousand

rupiah is twenty five thousand rupiah}


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So the reduction is Rp 25.000 {twenty five thousand rupiah}

Take the reduction from the original price

Rp100.000 – Rp 25.000 = Rp 75.000 (a hundred of thousand rupiah minus twenty

five thousand rupiah is seventy five thousand rupiah}

The Price of the Skateboard in the sale is Rp 75.000{seventy five thousand rupiah}

F. Percent vs Percentage

"Percentage" is the "result obtained by multiplying a quantity by a percent". So 10

{ten}percent of 50{fifty} apples is 5{five} apples: the 5{five} apples is the percentage.

But in practice people use both words the same way.


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Factoring in Algebra

A. Factors

Numbers have factors: {two times three is six}

And expressions (like x2+4x+3) {x square plus four x plus three} also have factors:

{bracket x plus three bracket x plus one

is x square plus four x plus three}

B. Factoring

Factoring (called "Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Example : factor 3y+9{three y plus nine}

Both 3y{three y} and 9{nine} have a common factor of 3{three}:

3y = 3 × y {three times y}

9 = 3 × 3 {three times three}

So you can factor the whole expression into:

3y+9 = 3(y+3) {three y plus nine is three bracket y plus three}

So, 3y+9 {three y plus nine}has been "factored into" 3{three} and y+3{y plus three}


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C. Common Factor

In the previous example we saw that 3y{three y} and 9{nine} had a common factor of 3

But, to make sure that you have done the job properly you need to make sure you have the

highest common factor, including any variables

Example: factor 3y2+12y {three y square plus twelve y}

Firstly, 3{three} and 12{twelve} have a common factor of 3{three}.

So you could have : 3y2+12y = 3(y2+4y) {three y square plus twelve y is three bracket y

square plus four y}

But we can do better!

3y2{three y square} and 12y{twelve y} also share the variable y.

Together that makes 3y{three y}:

3y2 = 3y × y {three y square is three y times three y}

12y = 3y × 4 {twelve y is three y times four}

So you can factor the whole expression into: 3y2+12y = 3y(y+4) {three y square plus twelve

y is three y bracket y plus four}

D. Experience Helps

The more experience you get, the easier it becomes.

Example: Factor 4x2 – 9 {four x square minus nine}


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But if you know your Special Binomial Products you might see it as the

"difference of squares":

Because 4x2 {four x square} is (2x)2 {bracket two x square}, and 9{nine} is (3)2{three

square},

so we have: 4x2 - 9 = (2x)2 - (3)2 {four x square minus nine is bracket two x square minus

three square}

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a2 - b2 {bracket a plus b bracket a minus b is a square minus b squar}

Where "a" is 2x{two x}, and "b" is 3{three}.

So let us try doing that:

(2x+3)(2x-3) = (2x)2 - (3)2 =, 4x2 - 9 {bracket two x, bracket two x minus three is bracket two

x, square minus three square is four x square minus nine}

So the factors of 4x2 – 9{four x square minus nine} are (2x+3){two x square plus three}

and (2x-3){two x plus three}:

Answer: 4x2 - 9 = (2x+3)(2x-3) {four x square minus nine is bracket two x plus three, bracket

two x minus three}

E. Advice

The factored form is usually best. When trying to factor, follow these steps:


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"Factor out" any common terms

See if it fits any of the identities, plus any more you may know

Keep going till you can't factor any more

F. Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

a2 - b2

{a square minus b square}=

(a+b)(a-b)Bracket a plus b, bracket a minus b

a2 + 2ab + b2

{a square plus two b plus bsquare}

=(a+b)(a+b)Bracket a plus b, bracket a plus b

a2 - 2ab + b2

a square minus two a b plus bsquare

=(a-b)(a-b)Bracket a minus b, bracket a minus b

a3 + b3

a cube plus b cube=

(a+b)(a2-ab+b2)Bracket a plus b, bracket a square minus two a b plusb square

a3 - b3 = (a-b)(a2+ab+b2)

a3+3a2b+3ab2+b3 = (a+b)3

a3-3a2b+3ab2-b3 = (a-b)3


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Angle

A. Definition of Angle

The angle is the set of two beam lines where the base of the two beam lines are allied

Given two intersecting lines or line segments, the amount of rotation about the point of

intersection (the vertex) required to bring one into correspondence with the other is called the

angle ߠ between them.

Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit),

or sometimes gradian (denoted grad).

The concept of an angle can be generalized from the circle to the sphere, in which case it is

known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is

measured in steradians, with the entire sphere corresponding to 4 {four phi} steradians.

B. Types of Angles

1. A full angle also called a perigon, is an angle equal to

2π {two phi} radians3600{three hundred sixty degree}.

It is corresponding to the central angle of an entire

circle. Four right angles or two straight angles equal one

full angle.

2. A reflex angle is an angle of more than 1800

{a hundred eighty degree}


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3. A straight angle is an angle equal to 0180 radians

4. A right angle is an angle equal to half the angle from

one end of a line segment to the other. A right angle is

{phi divide by two} radians or 900 {ninty degree}.

5. acute angle The angle of magnitude smaller than

900{ninety hundred} and bigger than the 00 {zero

hundred} ( 00 < α <900 ) {zero degree last than alpha last

than ninety degree}

C. Parts of an angel

1. The corner point of an angle is called the vertex

2. And the two straight sides are called arms


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Pythagoras’ Theorem

A. Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:

the square of the hypotenuse is equal to

the sum of the squares of the other two sides.

B. Formula of Pythagoras' Theorem

It is called "Pythagoras' Theorem" and can be written in one

short equation: a2 + b2 = c2

If we know the lengths of two sides of a right angled triangle,

we can find the length of the third side. (But remember it

only works on right angled triangles!) :

C. Example :

a2 + b2 = c2 {a square plus b square is c square}

52 + 122 = c2 {five square plus twelve square is c square}

25 + 144 = c2 {twenty five plus a hundred forty four is c square}

c2 = 169 {c square is a hundred sixty nine is c square}

c = √169 {c is square root of a hundred sixty nine}

c = 13 {c is thirteen}


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What is the diagonal distance across a square of size 1?


12 + 12 = c2 {one square plus one square is c square}

c2 = 2 {c square is two}

c = √2 = 1.4142... {c is square root of two =s one

point four one …..}


92 + b2 = 152 {nine square plus b square is fifteen square}

81 + b2 = 225 {eighty one plus b square is two hundred twenty

five}

b2 = 225 – 81{b square is two hundred twenty five minus eighty

one}

b2 = 144 {b square is a hundred forty four}b = √144 {b is square

root of a hundred forty four}

b = 12 {b is twelve}


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Exponents

A. Definition

Exponents are also called Powers or Indices

The exponent of a number says how many times to use

the number in a multiplication.

In this example: 82 = 8 × 8 = 64 {eight square is eight

times eight is sixty four}

In words: 82 {eight square} could be called "8

{eight} to the second power", "8{eight} to the

power 2{two}" or simply "8{eight} squared"

Example: 53 = 5 × 5 × 5 = 125 {five cubic is five times five times five is a

hundred twenty five}

In words: 53{five cubed} could be called "5{five} to the third power", "5{five} to the

power 3{three}" or simply "5{five} cubed"

Example: 24 = 2 × 2 × 2 × 2 = 16 {two power four is two times two times two

times two is sixteen}

In words: 24 {two power four} could be called "2{two} to the fourth power" or

"2{two} to the power 4{four}" or simply "2{two} to the 4th{fourth}"


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Exponents make it easier to write and use many multiplications

Example: 96 {nine power six} is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9 {nine

times nine times nine times nine times nine times nine}

You can multiply any number by itself as many times as you want using exponents.

B. In General

So, in general:

an tells you to multiply a by itself,

so there are n of those a's:

C. Other Way of Writing It

Sometimes people use the ^ symbol (just above the 6 on your keyboard), because it is easy to

type.

Example: 2^4 {two power four} is the same as 24{two power four}.

2^4 = 2 × 2 × 2 × 2 = 16 {two power four is two times two times two times two}.

D. Negative Exponents

A negative exponent means how many times to divide one by the number.

Example: 8-1 = 1 ÷ 8 = 0.125 { eight power minus one is one divide by eight is zero point a

hundred twenty five}.

You can have many divides:


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Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008 {five power minus three is one by five by five by five

s zero point zero zero eight}.

But that can be done an easier way:

5-3 {five power minus three} could also be calculated like:

1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008 {one divide by five times five times five is one divide

b five power three is one by a hundred twenty five}

E. In General

That last example showed an easier way to handle negative exponents:

Calculate the positive exponent (an)

Then take the Reciprocal (i.e. 1/an)

More Examples:

Negative

Exponent

Reciprocal of Positive

ExponentAnswer

4-2{four power

minus two}=

1 / 42 {one by four

power two}=

1/16 = 0.0625{ one by sixteen is zero

point zero six two five}

10-3 =1 / 103 {one by ten

cubed}=

1/1,000 = 0.001{ one by a thousand is

zero point zero zero one}

F. What if the Exponent is 1, or 0?

1. If the exponent is 1, then you just have the number itself (example 91 = 9)

2. If the exponent is 0, then you get 1 (example 90 = 1)

3. But what about 00 ? It could be either 1 or 0, and so people say it is "indeterminate".


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G. Be Careful About Grouping

To avoid confusion, use parentheses () in cases like this:

With () :(-2)2 = (-2) × (-2) = 4 {bracket minus two square is minus two

times minus two is four}

Without

() :

-22 = -(22) = - (2 × 2) = -4 {minus two square is minus bracket

two times two is minus four}

With () : (ab)2 = ab × ab {bracket ab square is ab times ab}

Without

() :

ab2 = a × (b)2 = a × b × b {ab square is a times b square is a

times b times b}


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Decimals

A. Definition

A decimal Number (based on the number 10) contains a Decimal Point. We sometimes say

"decimal" when we mean anything to do with our numbering system, but a "Decimal

Number" usually means there is a Decimal Point.

B. Place Value

To understand decimal numbers you must first know about place value.When we write

numbers, the position (or "place") of each number is important.

In the number 327{ Three Hundred Twenty Seven}:

the "7" {seven} is in the Units

position, meaning just

7{seven} (or 7{seven}

"1"s{first),

the "2"{two} is in the Tens

position meaning 2{two} tens (or

twenty),

and the "3"{three} is in the Hundreds position, meaning 3{three} hundreds.

As we move left, each position is 10{ten} times bigger! From Units, to Tens, to

Hundreds.

As we move left, each position is 10{ten} times bigger! From Units, to Tens, to

Hundreds


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But what if we continue past Units?

What is 10{ten} times smaller than Units?

1/10 ths (Tenths) are!

But we must first write a decimal point,

so we know exactly where the Units position is:

"three hundred twenty seven and four tenths"

C. Decimal Point

The decimal point is the most important part of a Decimal Number. It is exactly to the right

of the Units position. Without it, we would be lost ... and not know what each position meant.

Now we can continue with smaller and smaller values, from tenths, to hundredths, and so

on, like in this example:


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D. Large and Small

So, our Decimal System lets us write numbers as large or as small as we want, using the

decimal point. Numbers can be placed to the left or right of a decimal point, to indicate

values greater than one or less than one.

17.591

The number to the left of the decimal point is a whole

number (17 for example)

As we move further left, every number place gets 10 times bigger.

The first digit on the right means tenths (1/10).

As we move further right, every number place gets 10

times smaller (one tenth as big).

E. Ways to think about Decimal Numbers


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1. as a Whole Number Plus Tenths, Hundredths, etc

You could think of a decimal number as a whole number plus tenths, hundredths, etc:

Example 1: What is 2.3 {two point three}?

On the left side is "2"{two}, that is the whole number part.

The 3{three} is in the "tenths" position, meaning "3{three} tenths", or 3/10{three by

ten}

So, 2.3 {two point three} is "2{two} and 3{three} tenths"

Example 2: What is 13.76 {thirteen point seventy six}?

On the left side is "13"{thirteen}, that is the whole number part.

There are two digits on the right side, the 7{seven} is in the "tenths" position, and the

6{six} is the "hundredths" position

So, 13.76 {thirteen point seventy six} is "13{thirteen} and 7{Seven} tenths and 6

{six}hundredths"

2. as a Decimal Fraction

Or, you could think of a decimal number as a Decimal Fraction.

A Decimal Fraction is a fraction where the denominator (the bottom number) is a number

such as 10, 100, 1000, {ten, a hundred, a thousand}etc (in other words a power of ten)

So "2.3" would look like this:10

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And "13.76" would look like this:100

1376


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3. as a Whole Number and Decimal Fraction

Or, you could think of a decimal number as a Whole Number plus a Decimal Fraction.

So "2.3" would look like this: 2 and10

3

And "13.76" would look like this: 13 and100

76

Those are all good ways to think of decimal numbers.

A. Definition

In general, a cone is a pyramid

cone has a circular base and a vertex that is not on the base. Cones

are similar in some ways to pyramids. They both have just one

base and they converge to a point, the vertex.

B. Cone Facts

Notice these interesting things:

It has a flat base

It has one curved side

Because it has a curved surface it is not a

polyhedron.

And for reference:

Surface Area of Base = π

Surface Area of Side = π

Or Surface Area of Side = π

Volume = π × r2 × (h/3)


Cone

pyramid with a circular cross section. A

and a vertex that is not on the base. Cones

are similar in some ways to pyramids. They both have just one

base and they converge to a point, the vertex.

Notice these interesting things:

it has a curved surface it is not a

π × r2

π × r × s

π × r × √(r2+h2)


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The pointy end of a cone is

called the vertex or apex

The flat part is the base

An object shaped like a

cone is said to be conical

C. A Cone is a Rotated Triangle

A cone is made by rotating a triangle!

The triangle has to be a right angled triangle, and it gets rotated around one of its two short

sides. The side it rotates around is the axis of the cone.

D. Volume of a Cone vs Cylinder

The volume formulas for cones and cylinders are very similar:

The volume of a cylinder is: π × r2 × h {phi times r square times height}

The volume of a cone is:π × r2 × (h/3) {phi times r square times height by

three}

So, the only difference is that a cone's volume is one third (1/3) {one by three} of a cylinder's.

So, in future, order your ice creams in cylinders, not cones, you get 3{three} times more!

E. Different Shaped Cones

F. Example :

We will need to calculate the surface area and the volume of cone

Area of the cone is πrs {phi r s}

Area of the base is πr2 {phi r square}

SA is πrs + πr2 {phi r s plus phi r square}

{volume of cone is one by three phi r square times height}

Let's find the volume of this cone.

We can substitute the values into the volume formula. When we perform thecalculations, we find that the volume is 150.72 cubic centimeters.


Different Shaped Cones

We will need to calculate the surface area and the volume of cone

{phi r s}

{phi r square}

{phi r s plus phi r square}

{volume of cone is one by three phi r square times height}

Let's find the volume of this cone.

We can substitute the values into the volume formula. When we perform thecalculations, we find that the volume is 150.72 cubic centimeters.


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We can substitute the values into the volume formula. When we perform the


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Triangle

1. Definiton

Triangle is a flat build of 1800 {a hundred eighty degree} and the number

of corners formed by connecting the three points that do not line up in one area.

Types of Triangles:

a. Same Side of the Triangle

Triangle three sides the same length.

The length AB = BC = AC {AB equal BC equal

AC

∠A = ∠B = ∠C = 600 {angle a equal angle B

equal angle C is sixty degree}

∠A + ∠B + ∠C = 1800 {angle A plus angle B

plus angle C is a hundred eighty degree}

b. Isosceles

Triangle has two equal angles and two sides of the same.

The length AC = CB {AC equal BC}

The point ∠ A = ∠B {angle A equal angle B

∠A + ∠B + ∠C = 1800 {angle A plus angle B plus angle C

is a hundred eighty degree}

c. The elbows-angled triangle

Triangle in which one of its corners 900 {ninety degree}

A = 900 {a is ninety degree} A B

C


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d. Any triangle

- The three sides are not equal in

length (AB ≠ AC ≠ BC) {AB not equal AC not

equal BC}

- The three corners are not as large (∠ A ≠ ∠ B ≠

∠ C) {angle A not equal angle B not equal angle C}

- ∠ A + ∠ B + ∠ C = 1800 {angle A plus angle B

plus angle C is a hundred eighty degree}


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Source

1. http://www.mathsisfun.com

2. http://mathworld.wolfram.com

3. http://www.google.com/terjemahan

4. http://www.math.com/school/subject3/lessons/S3U4L4GL.html

5. http://www.mathsisfun.com/geometry/hexahedron.html

6. http://mathworld.wolfram.com/Cube.html

7. http://www.math.com/school/subject3/lessons/S3U4L4GL.html

8. http://math.about.com/od/formulas/ss/surfaceareavol_2.htm

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Book citra math 3_e