1

First of all, I would like to say Alhamdulillah, for giving me the

strength and health to do this project work.

Not forgotten my parents for providing everything, such as

money, to buy anything that are related to this project work and their

advise, 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 Rosmeela for guiding me and

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

task, but she taught us patiently until we knew 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.

2

The aims of carrying out this project work are:

i. to apply and adapt a variety of problem-solving strategies to

solve problems;

ii. to improve thinking skills;

iii. to promote effective mathematical communication;

iv. to develop mathematical knowledge through problem solving

in a way that increases students’ interest and confidence;

v. to use the language of mathematics to express mathematical

ideas precisely;

vi. to provide learning environment that stimulates and enhances

effective learning;

vii. to develop positive attitude towards mathematics.

3

Pi or π is a mathematical constant whose value is the ratio of any circle's

circumference to its diameter in Euclidean space; this is the same value as the

ratio of a circle's area to the square of its radius. It is approximately equal to

3.14159 in the usual decimal notation (see the table for its representation in

some other bases). π is one of the most important mathematical and physical

constants: many formulae from mathematics, science, and engineering involve

π.[1]

π is an irrational number, which means that its value cannot be expressed exactly

as a fraction m/n, where m and n are integers. Consequently, its decimal

representation never ends or repeats. It is also a transcendental number, which

means that no finite sequence of algebraic operations on integers (powers, roots,

sums, etc.) can be equal to its value; proving this was a late achievement in

mathematical history and a significant result of 19th century German

mathematics. Throughout the history of mathematics, there has been much effort

to determine π more accurately and to understand its nature; fascination with the

number has even carried over into non-mathematical culture.

The Greek letter π, often spelled out pi in text, was adopted for the number from

the Greek word for perimeter "περίµετρος", first by William Jones in 1707, and

popularized by Leonhard Euler in 1737.[2] The constant is occasionally also

referred to as the circular constant, Archimedes' constant (not to be confused

with an Archimedes number), or Ludolph's number (from a German

mathematician whose efforts to calculate more of its digits became famous).

4

5

6

The Letter Pi

The name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts

when the Greek letter is not available, or its usage could be problematic. It is not normally

capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is

always pronounced like "pie" in English, which is the conventional English pronunciation of the

Greek letter. In Greek, the name of this letter is pronounced /pi/.

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery)

and περίµετρος (perimeter), probably referring to its use in the formula to find the circumference, or

perimeter, of a circle.[3] π is Unicode character U+03C0 ("Greek small letter pi").[4]

Lower-case π is used to symbolize the constant.

Definition

Circumference = π × diameter

7

Area of the circle = π × area of the shaded square

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:[3]

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

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

Alternatively π 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 are

similar. This can be considered a problem when π occurs in areas of mathematics that otherwise do

not involve geometry. For this reason, mathematicians often prefer to define π without reference to

geometry, instead selecting one of its analytic properties as a definition. A common choice is to

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

(equivalent) definitions.

Irrationality and transcendence

Being an irrational number, π cannot be written as the ratio of two integers. This was proven in 1761

by Johann Heinrich Lambert.[3] In the 20th century, proofs were found that require no prerequisite

knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known.[7][8] A

somewhat earlier similar proof is by Mary Cartwright.[9]

Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This

means that there is no polynomial with rational coefficients of which π is a root.[10] An important

consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates

of all points that can be constructed with compass and straightedge are constructible numbers, it is

impossible to square the circle: that is, it is impossible to construct, using compass and straightedge

alone, a square whose area is equal to the area of a given circle.[11] This is historically significant,

for squaring a circle is one of the easily understood elementary geometry problems left to us from

antiquity; many amateurs in modern times have attempted to solve each of these problems, and their

efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by

the amateur involved.

Numerical value

The numerical value of π truncated to 50 decimal places is:[12]

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

See the links below and those at sequence A000796 in OEIS for more digits.

8

While the value of π has been computed to more than a trillion (1012) digits,[13] elementary

applications, such as calculating the circumference of a circle, will rarely require more than a dozen

decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate

the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated

to 39 decimal places is sufficient to compute the circumference of any circle that fits in the

observable universe to a precision comparable to the size of a hydrogen atom.[14][15]

Because π is an irrational number, its decimal expansion never ends and does not repeat. This infinite

sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few

centuries has been put into computing more digits and investigating the number's properties.[16]

Despite much analytical work, and supercomputer calculations that have determined over 1 trillion

digits of π, no simple base-10 pattern in the digits has ever been found.[17] Digits of π are available

on many web pages, and there is software for calculating π to billions of digits on any personal

computer.

Calculating π

π can be empirically estimated by drawing a large circle, then measuring its diameter and

circumference and dividing the circumference by the diameter. Another geometry-based approach,

due to Archimedes,[18] is to calculate the perimeter, Pn , of a regular polygon with n sides

circumscribed around a circle with diameter d. Then

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes

determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon

with the perimeter of a regular polygon with the same number of sides inscribed inside the circle.

Using a polygon with 96 sides, he computed the fractional range:

π can also be calculated using purely mathematical methods. Most formulae used for calculating the

value of π have desirable mathematical properties, but are difficult to understand without a

background in trigonometry and calculus. However, some are quite simple, such as this form of the

Gregory-Leibniz series:[20]

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In

addition, this series converges so slowly that 300 terms are not sufficient to calculate π correctly to 2

decimal places.[21] However, by computing this series in a somewhat more clever way by taking the

midpoints of partial sums, it can be made to converge much faster. Let

and then define

9

then computing π10,10 will take similar computation time to computing 150 terms of the original

series in a brute-force manner, and \pi_{10,10}=3.141592653\ldots, correct to 9 decimal places. This

computation is an example of the Van Wijngaarden transformation.[22]

History

The history of π parallels the development of mathematics as a whole.[23] Some authors divide

progress into three periods: the ancient period during which π was studied geometrically, the

classical era following the development of calculus in Europe around the 17th century, and the age of

digital computers.[24]

Geometrical period

That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is

slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The

earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81

(Egypt), both within 1% of the true value.[3] The Indian text Shatapatha Brahmana gives π as

339/108 ≈ 3.139. The Hebrew Bible appears to suggest, in the Book of Kings, that π = 3, which is

notably worse than other estimates available at the time of writing (600 BC). The interpretation of

the passage is disputed,[25][26] as some believe the ratio of 3:1 is of an interior circumference to an

exterior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the

thickness of the walls (See: Biblical value of π).

Archimedes (287-212 BC) was the first to estimate π rigorously. He realized that its magnitude can

be bounded from below and above by inscribing circles in regular polygons and calculating the outer

and inner polygons' respective perimeters:[26]

Liu Hui's π algorithm

10

By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 22/7.[26] Taking the

average of these values yields 3.1419.

In the following centuries further development took place in India and China. Around AD 265, the

Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate

π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained

an approximate value for π of 3.1416.

Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of

3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive

polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed

that 3.1415926 < π < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value was

the most accurate approximation of π available for the next 900 years.

Classical period

Until the second millennium, π was known to fewer than 10 decimal digits. The next major advance

in π studies came with the development of calculus, and in particular the discovery of infinite series

which in principle permit calculating π to any desired accuracy by adding sufficiently many terms.

Around 1400, Madhava of Sangamagrama found the first known such series:

This is now known as the Madhava-Leibniz series[27][28] or Gregory-Leibniz series since it was

rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of

convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to

improve upon Archimedes' estimate. However, by transforming the series into

Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was

beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who determined 16 decimals of π.

The first major European contribution since Archimedes was made by the German mathematician

Ludolph van Ceulen (1540–1610), who used a geometric method to compute 35 decimals of π. He

11

was so proud of the calculation, which required the greater part of his life, that he had the digits

engraved into his tombstone.[29]

Around the same time, the methods of calculus and determination of infinite series and products for

geometrical quantities began to emerge in Europe. The first such representation was the Viète's

formula,

found by François Viète in 1593. Another famous result is Wallis' product,

by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits,

although he later confessed: "I am ashamed to tell you to how many figures I carried these

computations, having no other business at the time."[30]

In 1706 John Machin was the first to compute 100 decimals of π, using the formula

with

Formulas of this type, now known as Machin-like formulas, were used to set several successive

records and remained the best known method for calculating π well into the age of computers. A

remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a

Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. The best value

at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707

digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern

record calculations of any kind are often performed twice, with two different formulas. If the results

are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved

through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761,

and Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735

solved the famous Basel problem – finding the exact value of

12

which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre

and Leonhard Euler speculated that π might be transcendental, which was finally proved in 1882 by

Ferdinand von Lindemann.

William Jones' book A New Introduction to Mathematics from 1706 is said to be the first use of the

Greek letter π for this constant, but the notation became particularly popular after Leonhard Euler

adopted it in 1737.[31] He wrote:

“ There are various other ways of finding the Lengths or Areas of particular Curve Lines, or

Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to

the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) + ... = 3.14159... = π[3] ”

Computation in the computer age

The advent of digital computers in the 20th century led to an increased rate of new π calculation

records. John von Neumann used ENIAC to compute 2037 digits of π in 1949, a calculation that took

70 hours. Additional thousands of decimal places were obtained in the following decades, with the

million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new

algorithms. One of the most significant developments was the discovery of the fast Fourier transform

(FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers

quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many

new formulas for π, some remarkable for their elegance and mathematical depth.[32] One of his

formulas is the series,

and the related one found by the Chudnovsky brothers in 1987,

which deliver 14 digits per term.[32] The Chudnovskys used this formula to set several π computing

records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691)

decimals in 1989. It remains the formula of choice for π calculating software that runs on personal

computers, as opposed to the supercomputers used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist

iterative algorithms that multiply the number of correct digits at each step, with the downside that

each step generally requires an expensive calculation. A breakthrough was made in 1975, when

Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which

uses only arithmetic to double the number of correct digits at each step.[33] The algorithm consists

of setting

13

and iterating

until an and bn are close enough. Then the estimate for π is given by

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that

quadruples the accuracy in each step has been found by Jonathan and Peter Borwein.[34] The

methods have been used by Yasumasa Kanada and team to set most of the π calculation records since

1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is

1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous

records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-

like formulas that were slower but crucially reduced memory consumption. The calculation was

performed on a 64-node Hitachi supercomputer with 1 terabyte of main memory, capable of carrying

out 2 trillion operations per second.

An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula),

discovered by Simon Plouffe and named after the authors of the paper in which the formula was first

published, David H. Bailey, Peter Borwein, and Plouffe.[35] The formula,

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without

calculating all the preceding ones.[35] Between 1998 and 2000, the distributed computing project

PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth

(1,000,000,000,000,000:th) bit of π, which turned out to be 0.[36]

In 2006, Simon Plouffe found a series of beautiful formulas.[37] Let q = eπ, then

and others of form,

14

where q = eπ, k is an odd number, and a,b,c are rational numbers. If k is of the form 4m+3, then the

formula has the particularly simple form,

for some rational number p where the denominator is a highly factorable number, though no rigorous

proof has yet been given.

Memorizing digits

Recent decades have seen a surge in the record for number of digits memorized.

Even long before computers have calculated π, memorizing a record number of digits became an

obsession for some people. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have

recited 100,000 decimal places.[38] This, however, has yet to be verified by Guinness World

Records. The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu

Chao, a 24-year-old graduate student from China.[39] It took him 24 hours and 4 minutes to recite to

the 67,890th decimal place of π without an error.[40]

There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a

piem, originally devised by Sir James Jeans: How I need (or: want) a drink, alcoholic in nature (or:

of course), after the heavy lectures (or: chapters) involving quantum mechanics.[41][42] Notice how

the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5,

and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner.[43] Piems are

related to the entire field of humorous yet serious study that involves the use of mnemonic

techniques to remember the digits of π, known as piphilology. In other languages there are similar

methods of memorization. However, this method proves inefficient for large memorizations of π. Other methods include remembering patterns in the numbers.[44]

Numerical approximations

Due to the transcendental nature of π, there are no closed form expressions for the number in terms

of algebraic numbers and functions.[10] Formulas for calculating π using elementary arithmetic

typically include series or summation notation (such as "..."), which indicates that the formula is

really a formula for an infinite sequence of approximations to π.[45] The more terms included in a

calculation, the closer to π the result will get.

Consequently, numerical calculations must use approximations of π. For many purposes, 3.14 or

22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6

significant figures) for more precision. The approximations 22/7 and 355/113, with 3 and 7

significant figures respectively, are obtained from the simple continued fraction expansion of π. The

approximation 355⁄113 (3.1415929…) is the best one that may be expressed with a three-digit or

four-digit numerator and denominator.[46][47][48]

The earliest numerical approximation of π is almost certainly the value 3.[26] In cases where little

precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the

fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

15

Open questions

The most pressing open question about π is whether it is a normal number—whether any digit block

occurs in the expansion of π just as often as one would statistically expect if the digits had been

produced completely "randomly", and that this is true in every base, not just base 10.[49] Current

knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur

infinitely often in the decimal expansion of π.[50]

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-

Plouffe formula and similar formulas imply that the normality in base 2 of π and various other

constants can be reduced to a plausible conjecture of chaos theory.[51]

It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved

the algebraic independence of {π, eπ, Γ(1/4)} in 1996.[52]

Use in mathematics and science

π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the

circles of Euclidean geometry.[53]

Geometry and trigonometry

For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr2. Further,

π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such

as ellipses, spheres, cones, and tori.[54] Accordingly, π appears in definite integrals that describe

circumference, area or volume of shapes generated by circles. In the basic case, half the area of the

unit disk is given by:[55]

and

gives half the circumference of the unit circle.[54] More complicated shapes can be integrated as

solids of revolution.[56]

From the unit-circle definition of the trigonometric functions also follows that the sine and cosine

have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn).

Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π

radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often defined using trigonometric functions, for example as the smallest

positive x for which sin x = 0, to avoid unnecessary dependence on the subtleties of Euclidean

geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions,

for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as

power series is the easiest way to derive infinite series for π.

16

Complex numbers and calculus

The frequent appearance of π in complex analysis can be related to the behavior of the exponential

function of a complex variable, described by Euler's formula

where i is the imaginary unit satisfying i2 = −1 and e ≈ 2.71828 is Euler's number. This formula

implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these

rotations have a period of 360° = 2π. In particular, the 180° rotation φ = π results in the remarkable

Euler's identity

There are n different n-th roots of unity

The Gaussian integral

A consequence is that the gamma function of a half-integer is a rational multiple of √π.

Physics

Although not a physical constant, π appears routinely in equations describing fundamental principles

of the Universe, due in no small part to its relationship to the nature of the circle and,

correspondingly, spherical coordinate systems. Using units such as Planck units can sometimes

eliminate π from formulae.

* The cosmological constant:[57]

* Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a

particle's position (∆x) and momentum (∆p) can not both be arbitrarily small at the same time:[58]

* Einstein's field equation of general relativity:[59]

17

* Coulomb's law for the electric force, describing the force between two electric charges (q1 and

q2) separated by distance r:[60]

* Magnetic permeability of free space:[61]

* Kepler's third law constant, relating the orbital period (P) and the semimajor axis (a) to the

masses (M and m) of two co-orbiting bodies:

Probability and statistics

In probability and statistics, there are many distributions whose formulas contain π, including:

* the probability density function for the normal distribution with mean µ and standard deviation

σ, due to the Gaussian integral:[62]

* the probability density function for the (standard) Cauchy distribution:[63]

Note that since for any probability density function f(x), the above formulas

can be used to produce other integral formulas for π.[64]

Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular

mathematics" works. Consider dropping a needle of length L repeatedly on a surface containing

parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it

comes to rest crossing a line (x > 0), then one may approximate π using the Monte Carlo

method:[65][66][67][68]

18

Though this result is mathematically impeccable, it cannot be used to determine more than very few

digits of π by experiment. Reliably getting just three digits (including the initial "3") right requires

millions of throws,[65] and the number of throws grows exponentially with the number of digits

desired. Furthermore, any error in the measurement of the lengths L and S will transfer directly to an

error in the approximated π. For example, a difference of a single atom in the length of a 10-

centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in

determining whether the needle actually crosses a line when it appears to exactly touch it will limit

the attainable accuracy to much less than 9 digits.

Pi in popular culture

Probably because of the simplicity of its definition, the concept of pi and, especially its decimal

expression, have become entrenched in popular culture to a degree far greater than almost any other

mathematical construct.[69] It is, perhaps, the most common ground between mathematicians and

non-mathematicians.[70] Reports on the latest, most-precise calculation of π (and related stunts) are

common news items.[71] Pi Day (March 14, from 3.14) is observed in many schools.[72] At least

one cheer at the Massachusetts Institute of Technology includes "3.14159!"[73] One can buy a "Pi

plate": a pie dish with both "π" and a decimal expression of it appearing on it.[74

19

20

QUESTION 2(a)

d1(cm) d2(cm) Length of arc PQR

in terms of π (cm)

Length of arc PAB

in terms of π (cm)

Length of arc BCR

in terms of π (cm)

1 9 5 π 0.5 π 4.5 π

2 8 5 π 1.0 π 4.0 π

3 7 5 π 1.5 π 3.5 π

4 6 5 π 2.0 π 3.0 π

5 5 5 π 2.5 π 2.5 π

6 4 5 π 3.0 π 2.0 π

7 3 5 π 3.5 π 1.5 π

8 2 5 π 4.0 π 1.0 π

9 1 5 π 4.5 π 0.5 π

10 0 5 π 5.0 π 0.0 π

Length of arc,

From the table, we can conclude that

Length of arc PQR = Length of arc PAB + Length of arc BCR

21

QUESTION 2(b)i

d1(cm) d2(cm) d3(cm)

Length of

arc PQR in

terms of π

(cm)

Length of

arc PAB in

terms of π

(cm)

Length of

arc BCD in

terms of π

(cm)

Length of

arc DER in

terms of π

(cm)

1 1 8 5 π 0.5 π 0.5 π 4.0 π

1 2 7 5 π 0.5 π 1.0 π 3.5 π

1 3 6 5 π 0.5 π 1.5 π 3.0 π

1 4 5 5 π 0.5 π 2.0 π 2.5 π

1 5 4 5 π 0.5 π 2.5 π 2.0 π

1 6 3 5 π 0.5 π 3.0 π 1.5 π

1 7 2 5 π 0.5 π 3.5 π 1.0 π

1 8 1 5 π 0.5 π 4.0 π 0.5 π

2 1 7 5 π 1.0 π 0.5 π 3.5 π

2 2 6 5 π 1.0 π 1.0 π 3.0 π

2 3 5 5 π 1.0 π 1.5 π 2.5 π

2 4 4 5 π 1.0 π 2.0 π 2.0 π

2 5 3 5 π 1.0 π 2.5 π 1.5 π

2 6 2 5 π 1.0 π 3.0 π 1.0 π

2 7 1 5 π 1.0 π 3.5 π 0.5 π

3 1 6 5 π 1.5 π 0.5 π 3.0 π

3 2 5 5 π 1.5 π 1.0 π 2.5 π

3 3 4 5 π 1.5 π 1.5 π 2.0 π

3 4 3 5 π 1.5 π 2.0 π 1.5 π

3 5 2 5 π 1.5 π 2.5 π 1.0 π

3 6 1 5 π 1.5 π 3.0 π 0.5 π

4 1 5 5 π 2.0 π 0.5 π 2.5 π

4 2 4 5 π 2.0 π 1.0 π 2.0 π

4 3 3 5 π 2.0 π 1.5 π 1.5 π

4 4 2 5 π 2.0 π 2.0 π 1.0 π

4 5 1 5 π 2.0 π 2.5 π 0.5 π

5 1 4 5 π 2.5 π 0.5 π 2.0 π

5 2 3 5 π 2.5 π 1.0 π 1.5 π

5 3 2 5 π 2.5 π 1.5 π 1.0 π

5 4 1 5 π 2.5 π 2.0 π 0.5 π

6 1 3 5 π 3.0 π 0.5 π 1.5 π

6 2 2 5 π 3.0 π 1.0 π 1.0 π

6 3 1 5 π 3.0 π 1.5 π 0.5 π

7 1 2 5 π 3.5 π 0.5 π 1.0 π

7 2 1 5 π 3.5 π 1.0 π 0.5 π

8 1 1 5 π 4.0 π 0.5 π 0.5 π

Length of arc,

22

From the table, we can conclude that

Length of arc PQR = Length of arc PAB + Length of arc BCD + Length of arc CDR

QUESTION 2(b)ii

Base on the findings in the table in(a) and (b) above, we conclude that:

The length of the arc of the outer semicircle = the sum of the

length of arcs of the inner semicircles for n inner semicircles

where n = 2, 3, 4…

Or

(s out) = ∑ n (s in), n = 2, 3, 4, ......

where,

s in = length of arc of inner semicircle

s out = length of arc of outer semicircle

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QUESTION 2(c)

Diagram above shows a big semicircle with n number of small inner circle. From the diagram, we

can see that

� � �1� �2 �… � �n

The length of arc of the outer semicircle

The sum of the length of arcs of the inner semicircles

Factorise π/2

Substitute

sout

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We get,

where d is any positive real number.

We can see that

EXAMPLES

c) Assume the diameter of outer semicircle is 30cm and 4 semicircles are inscribed in the outer

semicircle such that the sum of d1(APQ), d2(QRS), d3(STU), d4(UVC) is equal to 30cm.

d1 d2 d3 d4 SABC SAPQ SQRS SSTU SUVC

10 8 6 6 15π 5π 4π 3π 3π 12 3 5 10 15π 6 π 3/2 π 5/2 π 5 π 14 8 4 4 15π 7 π 4 π 2 π 2 π 15 5 3 7 15π 15/2 π 5/2 π 3/2 π 7/2 π

Let d1=10, d2=8, d3=6, d4=6, SABC= 5π + 4π + 3π + 3π

15 π = 5 π + 4 π + 3 π + 3 π 15 π = 15 π

The diameter of the outer semicircle, � � �1� �2�…� �n

10cm = 1cm + 1cm + 8cm

The length of arc of the outer semicircle, �1 + d2 + d3

0.5 π + 0.5 π + 4.0 π = 5 π

The sum of the length of arcs of the inner semicircles

Factorise π/2

(1cm + 1cm + 8cm) =5 π

In conclusion, we can conclude that

The length of the arc of the outer semicircle is equal to the sum of the length of

arcs of any number of the inner semicircles. This is true for any value of the

diameter of the semicircle.

In other words, for different values of diameters of the outer semicircle, the generalisations stated

in b (ii) is still true.

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26

27

QUESTION 3(a)

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QUESTION 3(b)

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QUESTION 3(c)

Linear Law

y = - π + π

Change it to linear form of Y = mX + C.

= -π +

Y =

X = x

m = -

C =

Thus, a graph of against x was plotted and the line of best fit was drewn.

X = x 1.0cm 1.5 cm 2.0 cm 2.5 cm 3.0 cm 3.5 cm 4.0 cm 4.5 cm 5.0 cm

Y = 7.069 6.676 6.283 5.890 5.498 5.105 4.712 4.320 3.927

Find the value of when x = 4.5 m.

Then multiply y/x you get with 4.5 to get the actual value of y.

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From the graph above, when the diameter of one of the fish pond is 4.5 m, the value of

is 4.35. Therefore, the area of the flower plot when the diameter of one of the fish

pond is 4.5 m is

4.35 m ( 4.5 m) = 19.575 m2

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Method 1: Differentation

y = - π + π

= -π + π

= - <--- y has maximum value

At maximum point, = 0

-π + π = 0

π = π

x = 5m

Maximum value of y = -52/4π + π

= 6.25π m2

Method 2: Completing the Square

y = - π + π

= - (x2 – 10x)

= - (x2 – 10x + 25 - 25)

= - [(x-5)2 – 25]

= - (x-5)2 + 25

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y is a n shape graph as a = -

Hence, it has a maximum value.

When x = 5 m, maximum value of the graph = 6.25π m2

QUESTION 3(e)

The principal suggested an additional of 12 semicircular flower beds to the design

submitted by the Mathematics Society. (n = 12)

The sum of the diameters of the semicircular flower beds is 10 m.

The diameter of the smallest flower bed is 30 cm. (a = 30 cm = 0.3 m)

The diameter of the flower beds are increased by a constant value successively. (d =

?)

S12 = ( )[2a + (n - 1)d]

10 = ( )[2(0.3) + (12-1)d]

= 6(0.6 + 11d)

= 3.6 + 66d

66d = 6.4

d =

Since the first flower bed is 0.3 m,

Hence the diameters of remaining 11 flower beds expressed in arithmetic

progression are:

m, m, m, m, m, m, m, m, m, m, m

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Part 1

Not all objects surrounding us are related to circles. If all the objects are circle, there

would be no balance and stability. In our daily life, we could related circles in objects.

For example: a fan, a ball or a wheel. In Pi(π), we accept 3.142 or 22/7 as the best

value of pi. The circumference of the circle is proportional as pi(π) x diameter. If the

circle has twice the diameter, d of another circle, thus the circumference, C will also

have twice of its value, where preserving the ratio =Cid

Part 2

The relation between the length of arcs PQR, PAB and BCR where the semicircles

PQR is the outer semicircle while inner semicircle PAB and BCR is Length of

arc=PQR = Length of PAB + Length of arc BCR. The length of arc for each

semicircles can be obtained as in length of arc = 1/2(2πr). As in conclusion, outer

semicircle is also equal to the inner semicircles where Sin= Sout .

Part 3

In semicircle ABC(the shaded region), and the two semicircles which is AEB and

BFC, the area of the shaded region semicircle ADC is written as in Area of shaded

region ADC =Area of ADC – (Area of AEB + Area of BFC). When we plot a straight

link graph based on linear law, we may still obtained a linear graph because Sin= Sout

where the diameter has a constant value for a semicircle.

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1. WIKIPEDIA

2. WWW.ONE-SCHOOL.NET

3. ADDITIONAL MATHEMATICS TEXTBOOK FORM 4 AND FORM 5