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    -Content-

    No. Contents Page

    1 Introduction 3 - 42 Objective 5

    3 Acknowledge 6

    4 Part 1 7 - 12

    5 Part 2 13 - 18

    7 Part 3 19- 23

    8 References 24

    -Introduction-

    A circle is a simple shape ofEuclidean geometry consisting of those points in a plane which

    are the same distance from a given point called thecentre. The common distance of the points of

    a circle from its center is called its radius. A diameteris a line segment whose endpoints lie on

    http://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpointhttp://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpoint
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    the circle and which passes through the centre of the circle. The length of a diameter is twice the

    length of the radius. A circle is never apolygon because it has no sides orvertices.

    Circles are simple closed curves which divide the plane into two regions, an interiorand an

    exterior. In everyday use the term "circle" may be used interchangeably to refer to either the

    boundary of the figure (known as theperimeter) or to the whole figure including its interior, but

    in strict technical usage "circle" refers to the perimeter while the interior of the circle is called a

    disk. The circumference of a circle is the perimeter of the circle (especially when referring to its

    length).

    A circle is a special ellipse in which the two foci are coincident. Circles are conic sections

    attained when a right circular cone is intersected with a plane perpendicular to the axis of the

    cone.

    The circle has been known since before the beginning of recorded history. It is the basis for

    the wheel, which, with related inventions such asgears, makes much of modern civilization

    possible. In mathematics, the study of the circle has helped inspire the development of geometry

    and calculus.

    Early science, particularly geometry and Astrology and astronomy, was connected to the divine

    for most medieval scholars, and many believed that there was something intrinsically "divine" or

    "perfect" that could be found in circles.

    Some highlights in the history of the circle are:

    1700 BC The Rhind papyrus gives a method to find the area of a circular field. The

    result corresponds to 256/81 as an approximate value of .[1]

    300 BC Book 3 ofEuclid's Elements deals with the properties of circles.

    1880 Lindemann proves that is transcendental, effectively settling the millennia-old

    problem ofsquaring the circle.[2]

    http://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Circle#cite_note-0http://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Circle#cite_note-0http://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-1
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    Objectives

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    The aims of carrying out this project work are:

    1. To apply and adapt a variety of problem-solving strategies to solve

    problems;

    2. To improve thinking skills;

    3. To promote effective mathematical communication;

    4. To develop mathematical knowledge through problem solving in a way

    that increase students interest and confidence;

    5. To use the language of mathematics to express mathematical ideas

    precisely;

    6. To provide learning environment that stimulates and enhances effective

    learning;

    7. To develop positive attitude toward mathematics.

    Acknowledgement

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    First of all I would like to say Alhamdulillah for giving the strength and

    health to do this project work.

    Not forgotten my parents for providing me everything such as money to

    buy anything that are related to this project work and their advice which is the

    most needed for this project. Internet , books , computers and all that . They also

    supported me and encouraged me to complete this task so that I will not

    procrastinate in doing it

    Then I would like to thank my teacher Mdm Asmalia Jaafar for guiding

    me and my friends throughout this project. We had some difficulties in doing this

    task , but she taught us patiently until we know what to do. She tried and tried

    to teach us until we understand what we supposed to do with the project work.

    Last but not least, my friends who were doing this project with me and

    sharing our ideas. They were helpful that when we combined and discussed

    together , we had this task done.

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    PART 1

    SYMBOLS

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    Wheel of a bicycle Circles on water surface School park

    Fish pond Round table at school compound

    Before I continue the task, first, we do have to know what dopi() related to a circle.

    Definition

    In Euclidean plane geometry, is defined as the ratio of a circle's circumference to its

    http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumference
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    diameter:

    The ratioC

    /d is constant, regardless of a circle's size. For example, if a circle has twice the

    diameterdof another circle it will also have twice the circumference C, preserving the ratio C/d.

    Area of the circle = area of the shaded square

    Alternatively can be also defined as the ratio of a circle's area (A) to the area of a square whose

    side is equal to the radius:[3][5]

    These definitions depend on results of Euclidean geometry, such

    as the fact that all circles aresimilar. This can be considered a

    problem when occurs in areas of mathematics that otherwise do

    not involve geometry. For this reason, mathematicians often prefer

    to define without reference to geometry, instead selecting one of

    itsanalytic properties as a definition. A common choice is to

    define as twice the smallest positivex for which cos(x) = 0.[6] The formulas below illustrate

    other (equivalent) definitions.

    History

    The ancient Babylonians calculated the area of a circle by taking 3 times the square of its

    radius, which gave a value ofpi = 3. One Babylonian tablet (ca. 19001680 BC) indicates a

    value of 3.125 forpi, which is a closer approximation.

    http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Pi#cite_note-adm-2http://en.wikipedia.org/wiki/Pi#cite_note-4http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-5http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Pi#cite_note-adm-2http://en.wikipedia.org/wiki/Pi#cite_note-4http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-5
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    P

    B

    C

    R

    Q

    A

    d1 d2

    10 cm

    PART 2

    Part 2 (a)

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    Diagram 1 shows a semicirclePQR of diameter 10cm. SemicirclesPAB andBCR of diameter d1

    and d2 respectively are inscribed inPQR such that the sum of d1 and d2 is equal to 10cm. By

    using various values of d1 and corresponding values of d2, I determine the relation between

    length of arcPQR,PAB, andBCR.

    Using formula: Arc of semicircle = d

    d1

    (cm)

    d2

    (cm)

    Length of arcPQR in

    terms of (cm)

    Length of arcPAB in

    terms of (cm)

    Length of arcBCR in

    terms of (cm)

    1 9 5 9/2

    2 8 5 4

    3 7 5 3/2 7/2

    4 6 5 2 3

    5 5 5 5/2 5/2

    6 4 5 3 2

    7 3 5 7/2 3/2

    8 2 5 4

    9 1 5 9/2

    Table 1

    From the Table 1 we know that the length of arcPQR is not affected by the different in d1 and d2

    inPAB andBCR respectively. The relation between the length of arcsPQR ,PAB andBCR is

    that the length of arcPQR is equal to the sum of the length of arcsPAB andBCR, which is we

    can get the equation:

    SPQR = SPAB + SBCR

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    P R

    Q

    A

    C

    d1 d2

    10

    cm

    Bd3

    D

    E

    Let d1= 3, and d2 =7 SPQR = SPAB + SBCR

    5 = (3) + (7)

    5 = 3/2 + 7/2

    5 = 10/2

    5 = 5

    The Arc length of sector

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    d1 d2 d3 SPQR SPAB SBCD SDER1 2 7 5 1/2 7/2

    2 2 6 5 3

    2 3 5 5 3/2 5/2

    2 4 4 5 2 2

    2 5 3 5 5/2 3/2

    SPQR = SPAB + SBCD + SDER

    Let d1 = 2, d2 = 5, d3 = 3 SPQR = SPAB + SBCD + SDER

    5 = + 5/2 + 3/2

    5 = 5

    The length of arc of sector

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    15 = 15

    The Arc length of sector

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    Part 3

    a. Area of flower plot = y m2

    y = (25/2) - (1/2(x/2)2 + 1/2((10-x )/2)2 )

    = (25/2) - (1/2(x/2)2 + 1/2((100-20x+x2)/4) )

    = (25/2) - (x2

    /8 + ((100 - 20x + x2

    )/8) )

    = (25/2) - (x2 + 100 20x + x2 )/8

    = (25/2) - ( 2x2 20x + 100)/8)

    = (25/2) - (( x2 10x + 50)/4)

    = (25/2 - (x2 - 10x + 50)/4)

    y = ((10x x2)/4)

    b. y = 16.5 m2

    16.5 = ((10x x2)/4)

    66 = (10x - x2) 22/7

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    2.0

    3.0

    4.0

    5.0

    6.0

    7.0

    8.0

    0 1 2 3 4 5 6 7

    X

    Y/x

    66(7/22) = 10x x2

    0 = x2 - 10x + 21

    0 = (x-7)(x 3)

    x = 7 , x = 3

    c. y = ((10x x2)/4)

    y/x = (10/4 - x/4)

    X 1 2 3 4 5 6 7

    y/x 7.1 6.3 5.5 4.7 3.9 3.1 2.4

    By using linear law method.

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    When x = 4.5 , y/x = 4.3

    Area of flower plot = y/x * x

    = 4.3 * 4.5

    = 19.35m2

    Thus, the area of flower plot is equal to 19.35m2

    d. Differentiation method

    dy/dx = ((10x-x2)/4)

    = ( 10/4 2x/4)

    0 = 5/2 x/2

    5/2 = x/2

    x = 5

    Completing square method

    y = ((10x x2)/4)

    = 5/2 - x2/4

    = -1/4 (x2 10x)

    y+ 52 = -1/4 (x 5)2

    y = -1/4 (x - 5)2 - 25

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    x 5 = 0

    x = 5

    e. n = 12, a = 30cm, S12 = 1000cm

    S12 = n/2 (2a + (n 1)d

    1000 = 12/2 ( 2(30) + (12 1)d)

    1000 = 6 ( 60 + 11d)

    1000 = 360 + 66d

    1000 360 = 66d

    640 = 66d

    d = 9.697

    Thus the diameter of the flower plot is 9.697cm

    Tn (flower bed) Diameter

    (cm)T1 30

    T2 39.697

    T3 49.394

    T4 59.091

    T5 68.788

    T6 78.485

    T7 88.182

    T8 97.879

    T9 107.576

    T10 117.273T11 126.97

    T12 136.667

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    REFERENCES

    Source via internet :

    http://en.wikipedia.org/wiki/Pi

    http://ualr.edu/lasmoller/pi.html

    http://egyptonline.tripod.com/history.htm

    http://www.gap-system.org/~history/HistTopics/Pi_through_the_ages.html

    News papers :

    News Straits Times

    Metro

    Berita Harian

    Magazines :

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    Siswa.

    Esti.my