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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