Area of Parallelogram

Parallelogram is a plane shape that has two pairs of parallel sides and equal in length. Parallelogram has two pairs of angles that are not right-angled. Opposite corners of a parallelogram having the same large angle. Parallelogram with four equal sides is called a rhombus.

The properties of a parallelogram is as follows:

• Opposite sides are parallel and equal in length
• Opposite angles are equal  
• Adjacent angles  add up to 180°, so they are supplementary angles (Two Angles are Supplementary if they add up to 180 degrees).
• Both diagonals bisect each line segment equal in length.

AC and BD are diagonal of ABCD parallelogram.

• AB equal length and parallel to the CD.
• BC equal length and parallel to AD.
• ∠ BAD = ∠ BCD, ∠ CBA = ∠ ADC
• OA = OB = OC and OD

Area of parallelogram can be found using triangle area by divide through BD or AC diagonals, so there are 2 congruent triangle. ΔBAD and ΔBCD.
Area of parallelogram = 2 x Area of triangle
                                       = 2 x ½ (b x h)
                                       = b x h
The Area is the base x height :
Area = b × h
Than from the area  of parallelogram can be found the base and heigh.
b  = A/h and h = A/b


Example :
1.  Height of a parallelogram is 15 cm and base = 25. What is the area ?
     Area = b x h
              = 15 x 25
              = 375 cm
2.  Area of parallelogram is 595 cm, height 17 cm. What is the base ?
     Base = Area/height
              = 595/17
              = 35 cm
3.  Base of the parallelogram is 16 cm and volume = 192 cm. What is the height ?
      Height = Area/Base
                   = 192/16
                   = 12 cm

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Reading and Writing Roman Numbers

There are various types of numbers, whole numbers ( whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on)) , counting numbers (Counting Numbers are Whole Numbers, but without the zero.), integers (Integers are like whole numbers, but they also include negative numbers), or fractions. One more type of numbers we will study the Roman numbers. In general, the Roman numbers consist of 7 digits (denoted by the letter) as follows.

• I represents the number 1
• V represents the number 5
• X represents the number 10
• L represents the number 50
• C represents the number 100
• D represents the number 500
• M represents the number 1000

For other numbers, denoted by a combination (mixture) of the seventh symbol number.

1. The sum rule of Roman Numbers
To read the Roman numbers, can we describe in the form of summation as in the example below.


Example:
a. II  = I + I
        = 1 + 1
        = 2
    So, II be read 2
b. VIII = V + I + I + I
            = 5 + 1 + 1 + 1
            = 8
     So, VIII be read 8
c. LXXVI = L + X + X + V + I
                 = 50 + 10 + 10 + 5 + 1
                 = 76
     So, LXXVI be read 76
d. CXXXVII = C + X + X + X + V + I + I
                     = 100 + 10 + 10 + 10 + 5 + 1 + 1
                      = 137
     So, read 137 CXXXVII
Try to note the number of Roman symbol of the examples above. Getting to the right, the smaller value. There is no symbol that lined the base of more than three. From these examples we can write the first rules in reading Roman numbers following the symbol.

• If the symbol of the smaller numbers ideally situated right, then the Roman symbols are summed. 
• The addition at most three points.

2. Reduction in Numbers of Roman rule

What if the symbol of the smaller number is located on the left? To read the Roman numbers, can we describe in the form of a reduction as in the example below.

Example:
a. IV = V - I
         = 5-1
         = 4
    Thus, IV to read 4
b. IX = X - I
          = 10-1
          = 9
    Thus, IX reads 9
c. XL = L - X
          = 50-10
          = 40
    So, read XL 40
From these examples we can write the second rule in reading Roman numbers following the symbol.

• If the symbol of the smaller number is located on the left, then the Roman symbols are deductible.
• Reduction of at most one point.

3. Joint Rules
The two rules above (addition and subtraction) can be combined so that it can more clearly in Roman numbers to read symbols. Let us consider the following example.
Example:
a. XIV = X + (V - I)
            = 10 + (5-1)
            = 10 + 4
            = 14
    So, XIV to read 14
b. MCMXCIX = M + (M - C) + (C - X) + (X - I)
                        = 1000 + (1000-100) + (100 -10) + (10-1)
                        = 1,000 + 900 + 90 + 9
                         = 1999
    So, read MCMXCIX 1999


Write Roman Numbers

After reading the Roman numbers, of course you can also write down the number of Roman symbol of the natural numbers are specified. Write the rules of the Roman symbol of the same number you have learned before. Let us consider the following example.

Example:
1.. 24 = 20 + 4
          = (10 + 10) + (5-1)
          = XX + IV
          = XXIV
    Thus, the symbol of the Roman number 24 is the XXIV
2. 48 = 40 + 8
         = (50 - 10) + (5 + 3)
         = XL + VIII
         = XLVIII
    Thus, the symbol of the Roman number 48 is XLVIII
3. 139 = 100 + 30 + 9
           = 100 + (10 + 10 + 10) + (10 - 1)
           = C + + XXX IX
           = CXXXIX
    Thus, the symbol of the Roman number 139 is CXXXIX
3. 1,496 = 1000 + 400 + 90 + 6
              = 1000 + (500-100) + (100-10) + (5 + 1)
              = M + CD + + XC VI
              = MCDXCVI
    Thus, the symbol of the Roman number 1496 is MCDXCVI

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Area of Trapezium/Trapezoid

A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides parallel.  Before looking trapezoid area, should know its parts. See the information below :

• The bottom side (base b ) parallel to the top side (base a) .
• The distance from one base to the other called height (h).

In the plane shapes are often written in capital letters at the vertex. This is to facilitate naming. For example, the trapezoid KLMN. KL means the line from  K point to L point, LM means the line from L point to M point, MN means the line from point M point to N point, KN means the line from K point to the N points.


Area of ​​trapezoid can be determined using the formula of triangle area. You do this by dividing the trapezoid
into two triangles. Then the triangle area summed.

Area of ​​trapezoid ABCD can be found by adding up the area of ​​triangle BAD with area of ​​triangle BCD

• BAD = ½ x AB x AD

                   = ½ x 6 x 5

                   =  ½ x 30

                   = 15 area unit

• BCD =  ½ x CD x BE

                   =  ½  x 9 x 5

                   =  ½  x 45

                   = 22,5 area unit

• ABCD = 15 + 22, 5

                      = 37, 5 area unit


Surface area of ​​the trapezoid = Area of ​​triangle I (BAD) + Area of ​​triangle  II (BCD)
                                                    =  ½ x a x h + ½ x bx h
                                                    = ½ (a+b) x h 

From the trapezoid area formula can be find high and the length of the trapezoid base.

• height = 2A/a + b
• a  base = (2A/h) - b
• b  base = (2A/h) - a

Example : 
1.  A trapezoid has 10 cm and 14 cm bases, height = 7 cm. What is the area ?
     Area = ½  x ( a + b) x h
               = ½ (10+14) x 7
               = ½ (24 ) x 7
               = 12 x 7
               =  84 cm
2.  A trapezoid has 8 cm and 12 cm bases, area = 90 cm. What is the height ?
      height = 2A/a + b
                  = 2(90)/8+12
                  = 180/20
                  = 9 cm
3.  A trapezoid has b base 16 cm, 14 cm height and area = 252 cm. What is the a base ?
     a  base = (2A/t) - b
                  = (2 x 252/14) - 16
                  = (504/14) - 16
                  = 36 - 16
                  = 20 cm
4. A trapezoid has a base 14 cm, 12 cm height and area = 180 cm. What is the b base ?
     b  base = (2A/t) - a
                  = (2 x 180/12) - 14
                  = (360/12) - 14
                  = 30 - 14
                  = 16 cm

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Area of Kite

You have to learn the triangle and the area of ​​the triangle. Isosceles triangles have special properties. Two isosceles triangles are the same length of its base can be compiled into a wake kites. Kites are rectangular. The kite has two pairs of sides of equal length. Kite is formed of two isosceles triangles. Both triangles have the same base length, but different height.  

Area of ​​kite can also be found using the formula area of ​​the triangle. By calculating the area of ​​the isosceles triangles that make up the kite. After that, the results are summed. Area of ​​kite ABCD can be found by summing the area ΔADC with ΔABC.

• Area  ΔADC =  ½ x  AC x OD

                                           = ½ x  8 x 4
                                           = 4 x 4
                                           = 16 area unit

• Area  ΔABC =  ½ x  AC x OB

                                           =  ½ x 8 x 9

                                           = 4 x 9

                                           = 36 area unit

• Area  ΔABC  = 16 + 36

                                = 52 area unit

Area ABCD = Area ΔADC + Area  ΔABC
                     = ½ x  AC x OD + ½ x  AC x OB
                            = ½ x AC ( OD + OB)
                            = ½ AC x BD 
                            = ½ x d1 x d2

 d1 and d2 is the diagonal of the kite. From the area of kite kite above, can be determined the diagonals.

• d1  = 2A/d2
• d2 = 2A/d1

Example :

1. A kite has d2 = 15 cm and area = 150 cm

     d1 = 2x150/15

           = 300/15

           = 20 cm

2.  A kite has d1 = 20 cm and area = 250 cm

      d2 = 2x 250/20

            = 500/20

            = 25 cm

3.  John want to make a kite. Two pieces of bamboo are made John with 48 cm and 44 cm 

      long.  If the kite has made, how area of the kite ?

      Area = ½ x d1 x d2

                 = ½ x 48 x 44

                 = 24 x 44

                 = 1.056 cm

4. On the wall there is a kite-shaped image. Area of the picture = 5.400 cm² and one of the 
    diagonal length is 120 cm. How long is the other diagonal ?
    d2 = 2A/d1
          = 2 x 5.400/120
          = 10.800/120
          = 90 cm

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Area of Triangle

Triangle is polygon ( a plane shape that have three or more straight sides) with three angles and three sides( One of the lines that make a flat (2-dimensional) shape. Or one of the surfaces that make a solid (3-dimensional) object). There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2 or no equal sides/angles:

Various kinds of triangles according to angle.

• Acute triangle . The third large angle of less than 90 °.
• Right triangle. One large angle of 90 °.
• Obtuse triangle. Big one corner more than 90 ° and less than 180 °

Various kinds of triangles according to the side.

• Scalene triangle, three sides of unequal length.
• Isosceles triangle, two sides equal in length.
• Equilateral triangle, three sides equal in length.

Equilateral Triangle =    Three equal sides  Three equal angles, always 60°

Isosceles Triangle = Two equal sides   Two equal angles

Scalene  Triangle = No equal sides   No equal angles

The characteristics of the triangle

• Triangles have three sides.
• Vertex of the triangle there are 3
• The whole  angles of triangle is 180 °.

Area of triangle

Area = ½ x b x h ( b=base, h=height)

Example :
Base of triangle : 15 cm and height : 12 cm
Area = ½ × b × h
         = ½ × 15 × 12
         = 90 cm

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Area of Rhombus

A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal. Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right angles. Rhombus formed from an isosceles triangle and its shadow reflected on the base as the axis of symmetry. Isosceles triangle ABC is reflected to the side of the base AC, so it appears his shadow ACD. ACD is congruent with the ABC. 

The properties of a rhombus as follows :

1. The length of its four sides equal in length 

     and Opposite sides are parallel.

• AB = BC = CD = AD
• AB // DC dan AD // BC

2. Both rhombus diagonal bisect each other 

    the same length and intersect 

    perpendicularly.

3. Opposite angles equal.

• ∠BAD =  ∠BCD
• ∠ABC =  ∠ADC
• ∠BAE =  ∠DAE =  ∠BCE = ∠ DCE
• ∠ADE =  ∠CDE =  ∠ABE =  ∠CBE

4. Both diagonal are  axis of symmetry.

• Diagonal AC ┘└ BD
• Panjang AE = EC
• Panjang DE = EB

Area of Rhombus :

Area =  ½ x d1 x d2
Based on rhombus area can be found each diagonal.

• d1  = 2A/d2
• d2 = 2A/d1

Example :

1. A rhombus has diagonal 1 = 15 cm and diagonal 2 = 20 cm. What is the area ?

    Area = di x d2

             = 15 x 20

             = 300 cm

2. A rhombus has 300 cm area and one of the diagonal is 30 cm. What is the other diagonal ?

     d1 = 2A/d2

          = 2(300)/30

          = 600/30

          = 20 cm

3.  A rhombus has 187 cm area and one of the diagonal is 17 cm. What is the other diagonal ?

     d2 = 2A/d1

          = 2(187)/17

          = 374/17

          = 22 cm

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Area of Circle

The circle is a plane shape that the distance of every point on its side with the center of the circle is always the same.. The radius of the circle is the distance of the center to the edge of the circle. The radius is denoted by r. The circle has a center line. The center line twice the length of the radius. The center line is denoted by diameter (d). d = 2 × r. The Circumference is the distance around the edge of the circle. Pi (the symbol is the Greek letter π) is: The ratio of the Circumference  to the Diameter of a Circle.

• π = circumference/diameter = 3,1415926535897323846....
• A quick and easy approximation to π is 22/7 = 3,1428571...

A circle with a center point C has the following sections.

• C is the center of the circle.
• AC = CB = r = radius of the circle.
• AB = diameter of the circle
• AB = AC + CB = 2 × r = 2R = 2 × radius
• If the radius of a circle = r and diameter d = 2 × r or r =  ½ × d

Area of ​​a circle is  π  times the square of the radius of the circle Area =

Circle Area = π r2

When the area of ​​a circle is expressed by the diameter :
Area = πr2  → (note: r = ½  x d)
          = π (½ d) 2
         = π ( ¼ d)2 
          = ¼ π d2 

Circle Area = ¼ π d2

Diameter = 2 x radius

Circumference =2πr or πd

Example :
1.  A circle has 21 cm radius. What is area and circumference ?
     Area = π r 2 
               = 22/7 x 212
               = 1.386 cm
    Circumference = 2πr
                               = 2 x 22/7 x 21
                               = 44 x 21
                               = 132 cm

2.  A circle has 616 cm area and π = 22/7. What is r  ?

     Area    =  π  x r2

     616     = 22/7 x r2

    22 x r2 = 616 x 7

    22r2     = 4312
        r2      = 4312/22
        r2     = 196
        r       =  √196
        r       = 14 cm

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How to Find the Prime Factors of a Number

A Prime Number can be divided evenly only by 1, or itself.  Example: 5 can only be divided evenly by 1 or 5, so it is a prime number. But 6 can be divided evenly by 1, 2, 3 and 6 so it is not a prime number (it is a composite number). A Composite Number can be divided evenly by numbers other than 1 or itself. Example: 9 can be divided evenly by 1, 3 and 9, so 9 is a composite number. But 7 can only be divided evenly by 1 and 7, so 7 is not a composite number (it is a Prime Number).  

If the integers p, p and p ¹ 0 dan p  ¹  1 only have 1 divider 1 and p, then p is called prime numbers. Integers other than primes are called composite numbers Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...etc. . Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, ...etc.

A factor that is a prime number. One of the prime numbers that, when multiplied, give the original number.
Example:

The prime factors of 35 are 5 and 7 (5×7=35, and 5 and 7 are prime numbers).

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Time Measurement Unit

As we all know, that the unit of time that there are various ranging from seconds, minutes, hours, days, weeks, months, years, and others. 1 hour = 60 minutes, 1 minute = 60 seconds to, 1 hour = 60 x 60 seconds = 3.600 seconds. 1 year = 12 months, 1 month = 30/31 days.

For example:
23 hours = ... minutes ?
23 hours = 23 x 60 minutes = 138 minutes.
43 minutes = ... second ?
43 minutes = 43 x 60 seconds = 258 seconds.
7 hours = ... second ?

7 hours = 7 x 60 x 60 seconds = 25 200 seconds. (View from hours to seconds the stairs down two times, so it must be multiplied by as much as 2 times the value 60 as well).

1 Week = 7 Days
1 Month = 28 to 31 Days
1 Month = 4 weeks
1 Semester = 6 Months
1 year = 365 to 366 Days
1 year = 52 weeks
1 year = 12 months
1 decade = 10 years
1 century = 100 years
1 millennium = 1,000 years

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Temperature Conversion

Temperature is a quantity that express the degree of heat and cold an object is a tool used to measure the temperature of the thermometer. In everyday life people tend to measure the temperature using the sense of touch. But with the development of the technology was created to measure the temperature with a thermometer valid.

The following table shows the temperature conversion formulas for conversions to and from the Celsius scale.

Example :
Temperature in the living room is 86⁰F . Calculate the temperature of living room in the C and R !

• C: R: F = 5: 4: 9,  F = 86 ⁰
• C/F= C = 5/9(86 ⁰  -32 ⁰ ) = 5/9 x 54 ⁰  = 212 ⁰ C
• R/F=4/9 =R = 4/9(86 ⁰  -32 ⁰ ) = 4/9 x 54 ⁰  = 24 ⁰ R
• So, 86 ⁰ F =30 ⁰ C =24 ⁰ R

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Calculating the Area of a Square

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90-degree angles, or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square is quadrilateral (Quadrilateral just means "four sides" quad means four, lateral means side).with :

• Four equal sides and four equal angles
• Opposite side that are parallel
• Two diagonal that bisect at right angles
• Four lines of symmetry

The area of a square can be found by multiplying the base times itself. This is similar to the area of a rectangle but the base is the same length as the height.
If a square has a side of length 7 cm its area is 7*7=49 square cm.

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Area of A Rectangle

A rectangle is a parallelogram with 4 right angles.  Now, since a rectangle is a parallelogram, its opposite sides must be congruent and it must satisfy all other properties of parallelograms .
1.  4 Right Angles, In a rectangle all angles are 90°

2. The diagonals of a rectangle are congruent (congruent means that objects have 

     the same shape. It does not mean that they are 'equal', exactly).

To find a diagonal using the Pythagorean theorem :
For example, a rectangle has a length of 12 cm and width 5 cm. How long the diagonal ?

The area of a rectangle can be found by multiplying the length times the width. The formula is: A= l * w, where  A is the area,  l is the length,  w is the width, and * means multiply. If a rectangle has a base of  length 10 cm and a width of 5 cm, its area is 10*5= 50 square cm :


Area = lenghth * width
          = 10 * 5
          = 50 cm

If the area of ​​the rectangle is known, to find the length and width can be done as follows:

#Note: one of the elements must be known, for example, the length element .

Area of Rectangle = 120 cm. What is the length ?

Length = Area/Width

              = 120/ 12

              = 10 cm

Do it the same way if the element width is already known

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Square Root of a Number

Square Root of a Number. Before understanding what the square root of a number is, it is important to understand the meaning of root of a number. The root of a number is an equal factor of the number. For example, find the root of 25

First, we need to factor 25 :
25 = 1 × 25
25 = 5 × 5
The root of 25 is 5 because 5 is the equal factor for 25
We call 5 the square root of 25
The symbol : √


How to find the square root :
1. Trial and error. 
This is the most common way to resolve the matter of the square root and is suitable for children who have been smoothly calculate the square or multiplication. For example we will calculate the square  roots 64.
a.  So we try 5 × 5 = 25 (too small).
b.  Try the 9 × 9 = 81 (too big).
c.  Try the 7 × 7 = 49 (too small).
d.  Try the 8 × 8 = 64 (right).    
     So we get the square root of 64 is 8.

2.  Factorization. 
This method is quite attractive and practical.  For example, what is the root of 64    
64 => 2 × 32
     => 2 x 2 x 16
     = 4 × 16
Then the root of 64 => root 4 x root 16                            
    => 2 x 4 => 8
3. Approaching

This method is a variation and the continuation of trial and error. After practicing a few times, we will be very proficient in this way.For example, what is the root of 1681?    

The most sensible approach is 40 × 40 = 1600.     

Because the unit value of 1681 is 1 so the roots of the unit 1 or 9.     

In this case we choose 1.  So we get the answer 40 +1 = 41  ( 41 x 41 = 1681)

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Volume and Surface Area of Triangular Prism

Volume and Surface Area of Triangular Prism. Triangular prism is an polyhedron. A prism with two identical bases. Prism is a solid object that has two identical ends and all flat sides. The cross section is the same all along its length. The shape of the ends give the prism a name, such as "triangular prism"

Surface Area of Triangular Prism
A = (b x h) + ( 3 x l x b )

Volume of Triangular Prism

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